A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs
Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, Takeaki Uno
AA POLYNOMIAL DELAY ALGORITHM FOR ENUMERATING MINIMALDOMINATING SETS IN CHORDAL GRAPHS
MAMADOU MOUSTAPHA KANTÉ, VINCENT LIMOUZY, ARNAUD MARY, LHOUARI NOURINE,AND TAKEAKI UNO
Abstract.
An output-polynomial algorithm for the listing of minimal dominating sets ingraphs is a challenging open problem and is known to be equivalent to the well-known Transver-sal problem which asks for an output-polynomial algorithm for listing the set of minimal hittingsets in hypergraphs. We give a polynomial delay algorithm to list the set of minimal dominatingsets in chordal graphs, an important and well-studied graph class where such an algorithm wasopen for a while. Introduction An enumeration algorithm for a set C is an algorithm that lists the elements of C withoutrepetitions. A hypergraph H is a pair p V, E q where V is a finite set and E Ď V is called the setof hyper-edges . Hypergraphs generalize graphs where each hyper-edge has size at most . Givena hypergraph H : “ p V, E q and C Ď V an output-polynomial algorithm for C is an enumerationalgorithm for C whose running time is bounded by a polynomial depending on the sum of thesizes of H and C . The enumeration of minimal or maximal subsets of vertices satisfying someproperty in a (hyper)graph is a central area in graph algorithms and for several propertiesoutput-polynomial algorithms have been proposed e.g. [2, 3, 11, 13, 27, 30, 32], while for othersit was proved that no output-polynomial algorithm exists unless P=NP [24, 25, 26, 27, 31].One of the central problem in the area of enumeration algorithm is the existence of an output-polynomial algorithm for the set of minimal transversals in hypergraphs, and is known as the Transversal problem or Hypergraph dualization . A minimal transversal (or hitting set ) in ahypergraph p V, E q is an inclusion-wise minimal subset T of V that intersects with every hyper-edge in E . The Transversal problem has several applications in artificial intelligence [10, 11],game theory [18, 29], databases [1, 5, 4, 33], integer linear programming [5, 4], to cite few.Despite the interest in the Transversal problem the best known algorithm is the quasi-polynomialtime algorithm by Fredman and Khachiyan which runs in time O p N log p N q q where N is thecumulated size of the given hypergraph and its set of minimal transversals. However, there existseveral classes of hypergraphs where an output-polynomial algorithm is known (see for instance[10, 11, 20] for some examples). Moreover, several particular subsets of vertices in graphs arespecial cases of transversals in hypergraphs and for some of them an output-polynomial algorithmis known, e.g. maximal independent sets, minimal vertex-covers, maximal (perfect) matchings,spanning trees, etc.In this paper we are interested in the particular case of the Transversal problem, namelythe enumeration of minimal dominating sets in graphs ( Dom-Enum problem). A minimaldominating set in a graph is an inclusion-wise subset D of the vertex set such that every vertexis either in D or has a neighbor in D . In other words D is a minimal dominating set of G if it is aminimal transversal of the closed neighborhoods of G . The closed neighborhood N r x s of a vertex x is the set containing x and its neighbors. Since in important graph classes an output-polynomialalgorithm for the Dom-Enum problem is a direct consequence of already tractable cases for theTransversal problem, e.g. minor-closed classes of graphs, graphs of bounded degree, it is naturalto ask whether an output-polynomial algorithm exists for the
Dom-Enum problem. However, itis proved in [20] that there exists an output-polynomial algorithm for the
Dom-Enum problem ifand only if there exists one for the Transversal problem, and this remains true even if we restrict
M.M. Kanté and V. Limouzy are supported by the French Agency for Research under the DORSO project. a r X i v : . [ c s . D S ] J u l M.M. KANTÉ, V. LIMOUZY, A. MARY, L. NOURINE, AND T. UNO the
Dom-Enum problem to the co-bipartite graphs. This is surprising, but has the advantageof bringing tools from graph structural theory to this difficult problem and is particularly truefor the
Dom-Enum problem since in several graph classes output-polynomial algorithms wereobtained using the structure of the graphs: graphs of bounded clique-width [6], split graphs[19, 20], interval and permutation graphs [22], line graphs [21, 23], etc.Since the
Dom-Enum problem in co-bipartite graphs is as difficult as the Transversal problemand co-bipartite graphs are a subclass of weakly chordal graphs, i.e. graphs with no cycles oflength greater than or equal to , one can ask whether by restricting ourselves to graphs withoutcycles of length , which are exactly chordal graphs [9], one cannot expect an output-polynomialalgorithm. In fact for several subclasses of chordal graphs an output-polynomial algorithm isalready known, e.g. undirected path graphs [21], split graphs, chordal P -free graphs [19, 20].Furthermore, chordal graphs have a nice structure, namely the well-known clique tree which hasbeen used to solve several algorithmic questions in chordal graphs. We prove the following. Theorem 1.
There exists a polynomial delay algorithm for the
Dom-Enum problem in chordalgraphs which uses polynomial space.
An output-polynomial algorithm is polynomial delay if the delay between two outputs isbounded by a polynomial in the size of the input (we also require the times before the firstoutput and after the last output to be bounded by polynomials on the input size). Amongoutput-polynomial algorithms polynomial-delay algorithms are the most desirable since theyallow to treat the solutions as they appear and we do not need to wait a long time between twooutputs. Notice that there exist problems where an output-polynomial algorithm is known andno polynomial delay algorithm exists unless P=NP [31].It is well-known that every n -vertex chordal graph G admits a linear ordering x , . . . , x n of itsvertex such that for every ď i ď n the vertex neighborhood of x i in G rt x i , . . . , x n us is a clique.For the enumeration of minimal dominating sets in chordal graphs the simplest strategy consistsin following this ordering as follows. Since N r x s is a clique, any minimal dominating set of G either contains x or does not contain x but contains at least one of its neighbors. Thereforeany minimal dominating set of G is either of the form D Yt x u where D is a minimal dominatingset of G z N r x s , or is a minimal dominating set of G zt x u that intersects the neighborhood of x .Unfortunately if the first case is just a recursive call, it is an exercice to see that the Transversalproblem reduces to the enumeration of minimal dominating sets of the second kind. Indeed,such a bottom-up strategy is hopeless since we will face the problem of identifying which setsin sub-trees are extendable to minimal dominating sets. An idea would be then to follow theclique tree in a top-down way, but as we will see if we do not take care we will come across an NP -complete problem. Our strategy will nevertheless follow the clique tree in a top-down way,but not in the usual way combining a kind of breadth-first search and depth-first search of thetree. We postpone the details of the strategy to forthcoming sections. Summary.
Definitions and preliminary results are given in Section 2. The strategy and somefaced difficulties are presented in Section 3. The algorithm and some necessary technical lemmasare given in Sections 4 and 5. We conclude with some open questions.2.
Preliminaries
General Definitions and Notations.
We refer to [8] for our graph terminology. Wedeal only with finite simple loopless undirected graphs. The vertex set of a graph G is denotedby V G and its edge set by E G . An edge between two vertices x and y is denoted by xy ( yx respectively). Let G be a graph. The subgraph of G induced by X Ď V G , denoted by G r X s isthe graph p X, p X ˆ X q X E G q . The size of a graph G , denoted by } G } , is | V G | ` | E G | , and the size of any C Ď V G , denoted by } C } , is defined as ř C P C | C | . For a vertex x of G we denote by N p x q the set of neighbors of x , i.e. the set t y P V G | xy P E G u , and we let N r x s , the closedneighborhood of x , be N p x q Y t x u . For S Ď V G , let N r S s denote Ť x P S N r x s . (We will removethe subscript when the graph is clear from the context and this will be the case for all sub orsuperscripts in the paper.) We say that a vertex x is dominated by a vertex y if x P N r y s . A OLY DELAY FOR
Dom-Enum
IN CHORDAL GRAPHS 3 dominating set of G is a subset D of V G such that every vertex of G is dominated by a vertex in D . A dominating set is minimal if it includes no other dominating set. For D Ď V G , a vertex x is a private neighbor of y P D if N r x s X D “ t y u ; the set of private neighbors of a vertex x P D is denoted by P p D, x q . D Ď V G is an irredundant set of G if P p D, x q ‰ H for all x P D . Thefollowing is easy to obtain. Fact 2. D Ď V G is a minimal dominating set of G if and only if D is a dominating set of G and D is an irredundant set. A clique of G is a subset C of G that induces a complete graph, and a maximal clique is aclique C of G such that C Y t x u is not a clique for all x P V G z C . We denote by C G the set ofmaximal cliques of G .A tree is an acyclic connected graph. Since we will talk at the same time about a graph anda tree representing it the vertices of trees will be called nodes . A rooted tree is a tree with adistinguished node, called its root , and let us denote by ĺ T the relation on a rooted tree T ,where u ĺ T v if v is on the unique path from the root to u ; if u ĺ T v then v is called an ancestor of u and u a descendant of v . Two nodes u and v of a rooted tree T are incomparable if u ĺ T v and v ĺ T u . Given a node u of a rooted T the subtree of T rooted at u is the tree T rt v P V T | v ĺ T u us which is rooted at u . A graph G is called chordal if it does not containchordless cycles of length greater than or equal to .Let G be a graph and let C be a subset of V G . An output-polynomial algorithm for C is an algorithm that lists the elements of C without repetitions in time O p p p} G } , } C }qq forsome polynomial p . We say that an algorithm enumerates C with polynomial delay if, after apre-processing that runs in time O p p p} G }qq for some polynomial p , the algorithm outputs theelements of C without repetitions, the delay between two consecutive outputs being boundedby O p q p} G }qq for some polynomial q (we also require that the time between the last outputand the termination of the algorithm is bounded by O p q p} H }qq ). It is worth noticing that analgorithm which enumerates a subset C of V G in polynomial delay outputs the set C in time O p p p} G }q ` q p} G }q ¨ | C | ` } C }q where p and q are respectively the polynomials bounding thepre-processing time and the delay between two consecutive outputs. Notice that any polynomialdelay algorithm is obviously an output-polynomial one, but not all output-polynomial algorithmsare polynomial delay [31]. We say that an output-polynomial algorithm uses polynomial spaceif there exists a polynomial q such that the space used by the algorithm is bounded by q p} G }q .2.2. Clique Trees of Chordal Graphs. An intersection graph is a graph in which each vertexcorresponds to a set and two vertices are adjacent if and only if their corresponding sets intersect.The collection of sets in correspondence with the vertices of an intersection graph is called an intersection model . Chordal graphs are exactly intersection graphs of subtrees in trees [15]. Achordal graph G admits at most | V G | maximal cliques. From [15] to every chordal graph G , onecan associate a tree that we denote by T G , called clique tree , whose nodes are in bijection withthe maximal cliques of G and such that for every vertex x P V G the set T G p x q : “ t u P V p T G q | the maximal clique of G corresponding to u contains x u is a subtree of T G . Moreover, G is theintersection graph of t T G p x q | x P V G u . Notice that there exist several clique trees for everychordal graph G , but we can compute one in linear time (see for instance [14]). Let us nowconsider some properties of clique trees. First of all, since the nodes of a clique tree T of G arein bijection with the maximal cliques of G each node of T will be identified with the maximalclique with which it is in correspondence. In the rest of the paper all trees are considered rooted.Let T G be a clique tree of a chordal graph G and let us denote its root by C r . For each C P C G ,let us denote by P a p C q its parent and let f p C q : “ C z P a p C q , i.e. , the set of vertices in C thatare not in any maximal clique C ancestor of C . Notice that t f p C q | C P T G u is a partitionof V G . For each vertex x P V G , we denote by C p x q the maximal clique C satisfying x P f p C q .Notice that C p x q is uniquely defined since exactly one maximal clique C satisfies x P f p C q . For C P C G , the subtree rooted at C is denoted by T G p C q , and the set of vertices Ť C P T G p C q f p C q isdenoted by V p C q . Property 3.
Any clique tree T G of a chordal graph G satisfies the following. M.M. KANTÉ, V. LIMOUZY, A. MARY, L. NOURINE, AND T. UNO (1)
For each C P C G , and each x P V G z V p C q either pt x uˆ f p C qq Ď E G or pt x uˆ f p C qqX E G “H . (2) For any two incomparable C and C in C G , we have p f p C q ˆ f p C qq X E G “ H . For S Ď V G let C p S q denote the set t C p x q | x P S u , U p p S q the set of vertices x in V G such that C p x q is a proper ancestor of a clique C P C p S q and U ncov p S q be the vertex set U p p S qz N r S s , i.e. the set of vertices in U p p S q not dominated by S . For a vertex x , U p p x q denotes U p pt x uq .A subset A Ď V G is an antichain if (1) for any two vertices x and y in A we have x R U p p y q and y R U p p x q , (2) for each vertex z P V G z U p p A q , A X p C p z q Y U p p z qq ‰ H . Intuitively, A is anantichain if C p A q is a maximal set of pairwise incomparable maximal cliques. Given S Ď V G , the top antichain A p S q is defined as the set of vertices of S included in the upmost cliques in C p S q that are not descendants of any other in C p S q , i.e., A p S q : “ t x P S | C p x q is in max ĺ T t C p S quu .If S ‰ H , let L p S q be the set of maximal cliques C satisfying (1) no descendant of C is in C p S q , (2) some descendants of P a p C q is in C p S q . In other words, L p S q is the set ofupmost maximal cliques no descendant of which intersects with C p S q , i.e. , L p S q : “ max ĺ T t C P C G | C has no descendant in C p S qu . If S “ H , let L p S q be t C r u . We denote by L p S q the set max ĺ T t C P T p C q | C P L p S q and C X S “ Hu .We suppose that any clique tree T is numbered by a pre-order of the visit of a depth-firstsearch. In this numbering, the numbers of the nodes in any subtree forms an interval of thenumbers. It is worth noticing that this ordering is a linear extension of the descendant-ancestorrelation. We say that a clique is smaller than another clique when its number is smaller thanthe other’s. We also extend this numbering to the vertices of the corresponding graph so thatthe number of a vertex x is smaller than that of a vertex y if C p x q is smaller than C p y q . Wealso say that a vertex is smaller than another vertex if its number is smaller than the other’s.For a vertex set S , tail p S q denote the largest vertex in S . A prefix of a vertex set S is its subset S such that no vertex in S z S is smaller than tail p S q . A partial antichain is a prefix of anantichain. We allow the H to be a partial antichain.Following this ordering of the vertices of a chordal graph G , a minimal dominating set D issaid to be greedily obtained if we initially let D : “ V G and recursively apply the following rule:if D is not minimal, find the smallest vertex x in D such that D zt x u is a dominating set and set D : “ D zt x u . Notice that given a graph G there is one greedily obtained minimal dominatingset. 3. When Simplicity Means NP-Hardness
A typical way for the enumeration of combinatorial objects is the back tracking technique.We start from the emptyset, and in each iteration, we choose an element x , and partition theproblem into two subproblems: the enumeration of those including x , and the enumeration ofthose not including x , and recursively solve these enumeration problems. If we can check the socalled Extension Problem in polynomial time, then the algorithm is polynomial delay anduses only polynomial space. The
Extension Problem is to answer the existence of an objectincluding S and that does not intersect with X , where S is the set (partial solution) that wehave already chosen in the ancestor iterations, and that includes all elements we decided to putin the output solution, and X is the set that we decided not to include in the output solution.It is known that the Extension Problem for minimal dominating set enumeration is NP -complete [28], and one can even prove that it is still NP -complete in split graphs (Proposition4), which are a proper subclass of chordal graphs. However, split graphs have a good structureand in the paper [19], it is proved that if S Y X induces a clique the Extension Problem insplit graphs can be solved in polynomial time and this combined with the structure of minimaldominating sets in split graphs lead to a polynomial delay algorithm for the
Dom-Enum problemin split graphs. Chordal graphs also have a good tree structure induced by clique trees. Thus,by following this tree structure, the
Extension Problem seems to be solvable. In precise,we consider the case in which a path P , from the root, of the clique tree satisfies that both V p C qXp S Y X q ‰ H and V p C q Ę p S Y X q holds only for cliques C included in P . In other words, OLY DELAY FOR
Dom-Enum
IN CHORDAL GRAPHS 5 the condition is that for any clique C R P whose parent is in P , either V p C q X p S Y X q “ H (totally not determined) or V p C q Ď p S Y X q (totally determined) holds. The solutions arepartially determined on the path P , and thus the Extension Problem seems to be polynomial.However, Theorem 5 states that the problem is actually NP -complete. Proposition 4.
The
Extension Problem is NP -complete in split graphs.Proof. It is proved in [28] that the following problem is NP -complete: Given G and A Ă V G decide whether there exists a minimal dominating set of G containing A . We reduce it to the Extension Problem in split graphs. Let G be a graph, and let V G : “ t x | x P V G u a disjointcopy of V G . We let Split p G q be the split graph with vertex set V G Y V G where V G and V G arerespectively the clique and the independent set in Split p G q ; now xy is an edge if x P N r y s . Nowit is easy to check that asking whether there exists a minimal dominating set of G that contains A Ă V G is equivalent to asking whether there exists a minimal dominating set of Split p G q thatcontains A and does not intersect with V G z A where A : “ N Split p G q r A s X V G . (cid:3) Theorem 5.
The
Extension Problem is NP -complete in chordal graphs even if a path P ,from the root, of the clique tree satisfies that any child C of a clique in P satisfies either V p C q Xp S Y X q “ H or V p C q Ď p S Y X q .Proof. We reduce
Sat to our problem. Let ϕ be an instance of Sat with x , . . . , x n the variablesand c , . . . , c m the clauses of ϕ . We construct a chordal graph as follows. The vertex set of thegraph is t x , . . . , x n , c , . . . , c m , p , . . . , p n , ¯ p , . . . , ¯ p n , l , . . . , l n u ď t ¯ l , . . . , ¯ l n , y , . . . , y n , z , . . . , z n , q , . . . , q n , ¯ q , . . . , ¯ q n u , where l i and ¯ l i are literals representing respectively x i and s x i (notice that if one literal does notappear, the corresponding vertex is not created). Since with every clique tree one can associate aunique chordal graph, we will construct the clique tree of the chordal graph. For each ď i ď n ,we let C p l i q and C p ¯ l i q be the set of clauses containing the literal l i and ¯ l i respectively. We letits root be C r : “ t c , . . . , c m , p , . . . , p n , ¯ p , . . . , ¯ p n u . The other maximal cliques are defined asfollows. For each ď i ď n , we let C x i “ t x i , p i , ¯ p i u , C y i “ t y i , x i u , C z i “ t y i , z i u , C q i “ t q i , l i u , C ¯ q i “ t ¯ q i , ¯ l i u , C l i “ t l i , p i u Y C p l i q , and C ¯ l i “ t ¯ l i , ¯ p i u Y C p ¯ l i q with the following parent-childrelation: C x i , C l i and C ¯ l i are the children of C r , C y i is the only child of C x i and C z i is the onlychild of C y i , C q i and C ¯ q i are the only children of C l i and C ¯ l i respectively. It is easy to checkthat the constructed tree is indeed a clique tree. See Figure 1 for an illustration.We set S : “ t x , . . . , x n , y , . . . , y n u and X : “ t z , . . . , z n , p , . . . , p n , ¯ p , . . . , ¯ p n uYt c , . . . , c m u and P : “ t C r u . For each ď i ď n , we have by construction V p C x i q Ď S Y X ,and p V p C l i q Y V p C ¯ l i qq X p S Y X q “ H . Therefore, for any maximal clique C child of C r , either V p C q X p S Y X q “ H , or V p C q Ď p S Y X q holds, thus the condition of the statement holds.One can easily check that any satisfiable assignment of ϕ leads to a minimal dominating setcontaining S and that does not intersect X . Let us prove the converse direction. We observewhen we choose both l i and ¯ l i in the dominating set, x i loses its private neighbors. Thus, anyminimal dominating set can include at most one of them. On the other hand, exactly one of l i and q i (resp., ¯ l i and ¯ q i ) must be included in any minimal dominating set, so that it dominates l i and q i (resp., ¯ l i and ¯ q i ), and both must be private neighbors of the chosen one. Moreover, todominate each clause c j , at least one literal of c j has to be included in any minimal dominatingset. Hence, for any minimal dominating set D including S and not intersect with X , the setof literals included in D corresponds to a satisfiable assignment. Therefore, the answer of the Extension Problem is yes if and only if ϕ has a satisfiable assignment. (cid:3) To overcome these difficulties, we will follow another approach. As we can see in the proof ofthe NP -completeness, when the root clique has both un-dominated vertices and private neighborsof several vertices of S , the Extension Problem turns to be difficult. In the following, wewill introduce a new strategy for the enumeration, that repeatedly enumerates antichains inlevelwise manner. Indeed for any minimal dominating set D of a chordal graph G the set A p D q M.M. KANTÉ, V. LIMOUZY, A. MARY, L. NOURINE, AND T. UNO C , . . . , C m p , . . . , p n p , . . . , p n C r x , p , p x , y y , z x n , p n , p n x n , y n y n , z n . . .. . .. . . | {z } Variables | {z }
Clauses l , p , C ( l ) l , q l n , p n , C ( l n ) l n , q n . . .. . . l , p , C ( l ) l , q l n , p n , C ( l n ) l n , q n . . .. . . Figure 1.
An illustration of the construction of Theorem 5.is an antichain that moreover dominates
U p p A p D qq . Our strategy consists in enumerating suchantichains and for each such antichain A enumerates the minimal dominating sets D such that A p D q “ A . Let’s be more precise in the next sections.4. p K , K q -Extensions Along this section we consider a fixed chordal graph G and clique tree T of G with root C r so that we do not need to recall them in the statements.Let K , K Ď C r be given disjoint sets that are decided to be included in the solution. Inour setting K will be the set of vertices that have already been assigned private neighborsso that we do not need to search one for them. Without confusion we denote K Y K by K . A p K , K q -extension of a partial antichain A is a vertex set D such that p A Y K q Ď D and D zp A Y K q Ď Ť C P L p A Y K q V p C q . Observe that if D is a p K , K q -extension of A , then A isa prefix of A p D q . When the partial antichain is not specified, p K , K q -extension is that forthe empty partial antichain. A p K , K q -extension D is feasible if it is a dominating set and P p D, x q ‰ H for all x P D z K . A partial antichain A is p K , K q -extendable if it has a feasible p K , K q -extension.For C P C G and x P C , let F p C, x q : “ t C ĺ T C and C P L p x qu , and let D C p x q denote avertex set composed of(1) Z Ď V p C q X ˜ Ť C P F p C,x q C ¸ such that | Z X C | “ | Z X f p C q| “ for all C P F p C, x q ,(2) a greedily obtained minimal dominating set of G rp V p C qz N r x sqz N r Z ss .If x R C , then we let D C p x q be a greedily obtained minimal dominating set of G r V p C qs . Property 6 (Irredundancy of D C p x q ) . Let C P C G and let x P V G . Then D C p x q is an irredun-dant set in G r V p C qs .Proof. Since each minimal dominating set is also an irredundant set, we can assume that x P C .By definition of Z we have that t x u ˆ Z X E G “ H . Moreover, by Property 3(2) no two verticesof Z are adjacent. Since by construction of D C p x qz Z no vertex in D C p x qz Z is adjacent to avertex of Z , we can conclude that for each z P Z we have z P P p D C p x q , z q . Moreover, since p D C p x qz Z q X N r Z s “ H and D C p x qz Z is a minimal dominating set of G rp V p C qz N r x sqz N r Z ss ,we can conclude that P p D C p x q , y q ‰ H for all y P p D C p x qz N r x sqz N r Z s . (cid:3) Property 7 (Domination of V p C qz N r x s ) . Let C P C G and let x P V G . Every vertex in V p C qz N r x s is dominated by D C p x q .Proof. If x R C , then D C p x q is a minimal dominating set of G r V p C qs and then we are done. So,assume that x P C and let y P V p C qz N r x s . Then C p y q is necessarily a descendant of a clique OLY DELAY FOR
Dom-Enum
IN CHORDAL GRAPHS 7 C P L p x q and such that C ĺ T C . So, either y P N r Z s or y R N r Z s . In both cases, it isdominated by D C p x q . (cid:3) Given disjoint sets K , K Ď C r , D Ď V G z K and x P D Y K , a vertex y P P p D Y K, x q issaid safe if either x “ y , or the following two conditions are satisfied(S1) N p y q X V p C q Ď N r D C p y qs for all C P L p D Y K q with y P C and(S2) for each z P N r y s X U ncov p D Y K q , there is a clique C P L p D Y K q such that z P N r D C p y qs .A vertex x P D is said safe if one of its private neighbors is safe. Property 8 (Domination of V p C qzt y u for safe y ) . Let x P D Y K and let y P P p D Y K, x q bea safe for x . Then V p C qzt y u Ď N r D C p y qs for all C P L p D Y K q with y P C .Proof. By Property 7 V p C qz N r y s is dominated by D C p y q . By definition of safety N p y q is dom-inated by D C p y q . Therefore V p C qzt y u is dominated by D C p y q for all C P L p D Y K q with y P C . (cid:3) We will now prove some technical lemmas that will be used to prove the correctness of thealgorithm.
Lemma 9 (Extension Safe) . Let A be a partial antichain and let x P A Y K . For y P P p A Y K, x q that is non-safe, no p K , K q -extension D of A that is a dominating set satisfies that y P P p D, x q .Proof. Since y is not safe, we have x ‰ y , and therefore y violates one of the two conditions (S1)or (S2) to be safe. Suppose that (S1) is not satisfied, i.e. there is a clique C P L p A Y K q , y P C such that there is a vertex z in p N p y q X V p C qqz N r D C p y qs . Thus, any p K , K q -extension D of A that is a dominating set includes some vertices in N r y s other than x , thus y is not a privateneighbor of x .Suppose now that (S2) is not satisfied, i.e. there is a vertex z P N r y s X U ncov p A Y K q suchthat no clique C P L p A Y K q satisfies z P N r D C p y qs . It implies from the definition of D C p y q that no vertex in V p C qz N r y s is adjacent to z in all cliques C P L p A Y K q . Thus, as in theprevious case, in any p K , K q -extension D of A , y is not a private neighbor of x unless D is nota dominating set. (cid:3) Lemma 10 (Lower Private Neighbor) . Let A be a partial antichain and let x P A Y K be safe.Then there is y P P p A Y K, x q that is safe and such that y P V p C p x qq .Proof. The statement holds if x P P p A Y K, x q . If not, C p x q includes another vertex in A Y K ,and it is adjacent to any vertex in N r x sz V p C p x qq by Property 3. Thus all its safe privateneighbors are always in V p C p x qq . (cid:3) Lemma 11 (Extendability of Partial Antichain) . A partial antichain A is p K , K q -extendableif and only if the following two conditions are satisfied (1) any vertex in U ncov p A Y K q is included in a clique of L p A Y K q , (2) all vertices in A Y K are safe.Proof. Let A be a p K , K q -extendable partial antichain. If (1) is not satisfied, there is a vertex z P U ncov p A Y K q that is not included in any clique of L p A Y K q , and by definition of p K , K q -extension no p K , K q -extension of A can dominate it. So (1) is always satisfied. Now, if (2) isnot satisfied, there is a non-safe vertex x in A Y K , thus all y P P p A Y K, x q are non-safe. ByLemma 9 it follows that P p D, x q “ H for each p K , K q -extension D of A that is a dominatingset, and then (2) is always satisfied.Suppose now that the two conditions hold. For each x P A Y K let us choose one safeprivate neighbor and let us denote the set of all these safe private neighbors by S . We considera p K , K q -extension D generated from A Y K as follows. First of all notice that from thedefinition of private neighbor and safety for each C P L p A Y K q , | C X S | ď . So, let L : “ t C P L p A Y K q | | C X S | “ u and L : “ t C P L p A Y K q | | C X S | “ u . It is clear that t L , L u is M.M. KANTÉ, V. LIMOUZY, A. MARY, L. NOURINE, AND T. UNO a bipartition of L p A Y K q . Let z P S . Now let D : “ p A Y K q Y ¨˝ ď C P L ,C X S “t y u D C p y q ˛‚ Y ˜ ď C P L D C p z q ¸ .D is clearly a p K , K q -extension of A . By definition of D C p y q for each vertex x P D zp A Y K q we have that P p D, x q ‰ H . It is moreover easy to check that for each x P A Y K , we have that S X P p A Y K, x q P P p D, x q . Thus, from Property 3, P p D, x q ‰ H for all x P D z K . Each vertexin N r A Y K s is dominated. Moreover, since for each C P L we have z R C , by definition of D C p z q we have V p C q is also dominated. Now, let C P L and let C X S “ t y u . We know fromProperty 7 that V p C qz N r y s is dominated by D C p y q and y is dominated by A Y K since y issafe for some vertex in A Y K . So, it remains to show that N p y q X V p C q is dominated. By thedefinition of safety we know that the two conditions (S1) and (S2) are satisfied, i.e. N p y qX V p C q is dominated. (cid:3) As a corollary we have the following.
Lemma 12.
For any partial antichain A one can check in polynomial time whether A is p K , K q -extendable.Proof. By Lemma 11 it is enough to check if (1) all vertices in A Y K are safe and (2) eachvertex in U ncov p A Y K q is included in a clique in L p A Y K q . Since (2) can be easily checkedin polynomial time from G and a clique tree of G , it remains to show that (1) can be checkedin polynomial time. A vertex x P A Y K is safe if either x P P p A Y K , x q or there existsa safe y P V p C p x qq X P p A Y K , x q by Lemma 10. But by the definition of safety for each y P V p C p x qq X P p A Y K , x q the conditions (S1) and (S2) are of course checkable in polynomialtime from G and a clique tree of G . (cid:3) The Algorithm
As in the previous section let us assume we are given a chordal graph G and a clique tree T of G rooted at C r . Remind that for a subset S of V G the top antichain of S denoted by A p S q is theset of vertices of S included in the upmost cliques in C p S q that are not descendants of any other in C p S q , i.e., A p S q : “ t v P S | C p v q is in max ĺ T t C p S quu . We observe that for any minimal dominatingset D of G , its top antichain is an pH , Hq -extendable antichain. Moreover, D z A p D q is composedof vertices below A p D q , i.e., any vertex in D z A p D q is included in V p C qz C for some C P C p D q .Using this, we partition the minimal dominating sets according to their top antichains. Sincethese top antichains are pH , Hq -extendable, we enumerate all pH , Hq -extendable antichains,and for each pH , Hq -extendable antichain A , enumerate all minimal dominating sets whose topantichain is A . As by definition of p K , K q -extendable for some disjoint K , K Ď C r , foreach pH , Hq -antichain A there is at least one minimal dominating set whose top antichain is A .Therefore, each output pH , Hq -antichain will give rise to a solution. This is one of the key topolynomial delay.Now for a minimal dominating set D and a clique C P C p A p D qq , each vertex x in D X p V p C q Y C q cannot have a private neighbor in another G r V p C q Y C s for some other C P C p A p D qq .Therefore, we can treat each G r V p C q Y C s independently. However, for each C P C p A p D qq theset D X p V p C q Y C q is not necessarily a minimal dominating set of G r V p C q Y C s since D X C may be equal to a singleton t x u with x having a private neighbor in U p p A p D qq . In such caseswe are looking in G r V p C q Y C s a dominating set D of G r V p C q Y C s containing x where x does not necessarily have a private neighbor, but all the other vertices in D do, i.e. D is afeasible pt x u , Hq -extension in G r V p C qY C s with clique tree T p C q . This situation is what exactlymotivated the notion of p K , K q -extensions.Assume now we are given a pair p K , K q of disjoint sets in C r and a p K , K q -extendableantichain A . Now contrary to pH , Hq -antichains we can have a vertex x in K : “ K Y K thatbelongs to several cliques in A . So we cannot independently make recursive calls in G r V p C q Y C s for each C P C p A q . But, for each feasible p K , K q -extension of A and each C P C p A q the set OLY DELAY FOR
Dom-Enum
IN CHORDAL GRAPHS 9 D X p V p C q Y C q is a feasible p K C , K C q -extension of G r V p C q Y C s for some disjoint K C and K C in p A Y K q X C . Now the whole task is to define for each C P C p A q the sets K C and K C in p A Y K q X C in such a way that by combining all these feasible p K C , K C q -extensionswe obtain a feasible p K , K q -extension of A , and also any feasible p K , K q -extension can beobtained in that way. Actually, the way of setting K C and K C is the key, and is described below.After defining K C and K C we will be able to enumerate all the feasible p K C , K C q -extensions in G r V p C q Y C s in the same way.In summary, our enumeration strategy is composed of nested enumerations: enumeration of p K , K q -extendable antichains, for each p K , K q -extendable antichain A and each C P C p A q define K C and K C and enumerate all the feasible p K C , K C q -extensions, and finally the combina-tions of all these p K C , K C q -extensions. Since any minimal dominating set is a feasible extensionof some pH , Hq -extendable antichain, the completeness of the enumeration is trivial. The restof the section is as follows. We first show how to enumerate p K , K q -extendable antichains forsome fixed p K , K q . Then we show, given a p K , K q -extendable antichain A , how to define K C and K C for each C P C p A q and how to combine all the feasible p K C , K C q -extensions in order toobtain all feasible p K , K q -extensions of A . Before assuming that we can perform both taskswith polynomial delay and use only polynomial space let us show that we can enumerate withpolynomial delay and use polynomial space all the feasible p K , K q -extensions.5.1. Enumeration of p K , K q -Extensions. This subsection deals with the algorithm for enu-merating all the feasible p K , K q -extensions, including the case of the root of the recursion. Aswe explained, the algorithm is composed of p K , K q -extendable antichain enumeration and ofthe enumeration of combinations of the feasible p K C , K C q -extensions for appropriate p K C , K C q .It can be described as follows. Algorithm
EnumKExtension p G, T , K , K q G :graph, T :clique tree, K , K : disjoint subsets of C r root of T for each antichain A output by EnumAntichain p G, T , K , K , Hq do output each solution of EnumCombination p G, T , K , K , A, A Y K q end for Assume that
EnumAntichain p G, T , K , K , Hq enumerates all p K , K q -extendable antichains(Lemma 14) and EnumCombination p G, T , K , K , A, A Y K q enumerates all feasible p K , K q -extensions of A (Lemma 17), both with polynomial delay and use polynomial space. Then wehave the following. Theorem 13.
The call
EnumKExtension p G, T , K , K q enumerates all feasible p K , K q -extensionsin polynomial delay and uses polynomial space.Proof. By definition for every feasible p K , K q -extension D the top antichain A p D z K q is a p K , K q -extendable antichain. So by Lemmas 14 and 17 below every feasible p K , K q -extensionis output. Therefore, EnumKExtension p G, T , K , K q enumerates all feasible p K , K q -extensions.From the definition of p K , K q -extendable antichains every call in Step 1 outputs at least onefeasible p K , K q -extension. Now since EnumAntichain p G, T , K , K , Hq and EnumCombination p G, T , K , K , A, A Y K q runs with polynomial delay and use both polynomialspace we can conclude that EnumKExtension p G, T , K , K q runs with polynomial delay and usepolynomial space. (cid:3) Enumeration of Antichains.
Our strategy is to enumerate all p K , K q -extendable par-tial antichains by an ordinary backtracking algorithm, that repeatedly appends a vertex to thecurrent solution that is larger than its tail. In this algorithm, any p K , K q -extendable partial an-tichain A is obtained from A z tail p A q . Since A z tail p A q is a prefix of A , any p K , K q -extendablepartial antichain is generated from another p K , K q -extendable partial antichain. This impliesthat the set of p K , K q -extendable partial antichains satisfies a kind of monotone property, andthus we can enumerate all p K , K q -extendable partial antichains with passing through only p K , K q -extendable partial antichains. The algorithm is described as follows. Algorithm
EnumAntichain p G, T , K , K , A q G :graph, T :clique tree, K , K : disjoint subsets of C r root of T A : p K , K q -extendable partial antichain1. if A is an antichain then output A ;2. for each vertex z ą tail p A q do if A Y t z u is a p K , K q -extendable partial antichain thencall EnumMinAntichain p G, T , K , K , A Y t z uq end for Lemma 14.
The call
EnumAntichain p G, T , K , K , Hq enumerates all p K , K q -extendable an-tichains in polynomial delay with polynomial space.Proof. We observe that for any p K , K q -extendable partial antichain A , A z tail p A q is a p K , K q -extendable partial antichain. Thus, one can easily prove by induction that the iteration inputting A is recursively called only by the iteration inputting A z tail p A q . Therefore, all p K , K q -extendable partial antichains are generated by this algorithm without repetition. For a p K , K q -extendable partial antichain A , there is at least one feasible p K , K q -extension D . By the defi-nition of a feasible p K , K q -extension, A p D z K q is a p K , K q -extendable antichain with A as aprefix. This implies that at least one descendant of any iteration outputs an antichain, and everyleaf of the recursion tree outputs an antichain. Then, the delay is bounded by the maximumcomputation time of an iteration multiplied by the depth of the recursion. The depth is at most | V G | , thus the algorithm is polynomial delay since the loop at Step 2 runs at most n times andthe p K , K q -extendability check can be done in polynomial time by Lemma 12. Since the depthis bounded by | V G | , the algorithm uses obviously a polynomial space. (cid:3) Enumeration of Combinations.
We now show, given a p K , K q -extendable antichain A , how to enumerate with polynomial delay and use only polynomial space all feasible p K , K q -extensions of A by computing for each C P C p A q all the p K C , K C q -extensions of G r V p C q Y C s for appropriate K C and K C and combine all of them. Note that the set A is the top antichain ofany feasible p K , K q -extension if and only if if the p K , K q -extension is that of A . For pruningredundant partial combinations, we introduce the notion of a partial p K , K q -extension. Avertex set D Ě A Y K is called a partial p K , K q -extension of A if there is a feasible p K , K q -extension D of A such that D zp A Y K q is a prefix of D zp A Y K q , and all the vertices in V p C p x qq for x P A is dominated by D if x is smaller than tail p D zp A Y K qq . Our strategy isto enumerate all partial p K , K q -extensions of A , similar to the antichain enumeration. Fora partial p K , K q -extension D of A , let C ˚ p D q be the largest clique C in C p A q such that p D zp A Y K qq X V p C q ‰ H , and C ˚ p D q be the smallest clique C in C p A q such that a vertex in V p C q is not dominated by D . Informally C ˚ p D q is the last clique C P C p A q such that V p C q isdominated by D , and C ˚ p D q the first clique in C p A q such that V p C q is not dominated by D . Toenumerate all partial p K , K q -extensions of A and in fine all p K , K q -extensions of A , we startfrom D “ A Y K and repeatedly add a p K C ˚ p D q , K C ˚ p D q q -extension of G r V p C ˚ p D qq Y C ˚ p D qs to D for appropriate p K C ˚ p D q , K C ˚ p D q q , while keeping the extendability. To characterize thepossible p K C ˚ p D q , K C ˚ p D q q we state the following lemma. Let Q D p C q be the vertices x in K Y A that have no safe private neighbor in V p C q Y C, C ą C , and none of its private neighbor in P p K Y A Y D, x q is included in U p p A qz C or in V p C q , C ă C . In other words Q D p C q is the set ofvertices in K Y A that we must give a private neighbor in V p C q Y C for any p K , K q -extensionof A containing D . Lemma 15.
For a non-empty partial p K , K q -extension D , D X p V p C ˚ p D qq Y C ˚ p D qq is afeasible p K , K q -extension in G r V p C ˚ p D qq Y C ˚ p D qs where K “ Q D p C ˚ p D qq and K “ pp A Y K q X C ˚ p D qqz K .Proof. By definitions of partial p K , K q -extension and of C ˚ , D Xp V p C ˚ p D qY C ˚ p D qq dominates V p C ˚ p D qq . Moreover, every vertex x in Q D p C ˚ p D qq has a private neighbor only in V p C ˚ p D qq Y C ˚ p D q , and moreover x P C ˚ p D q . Thus, the statement holds. (cid:3) OLY DELAY FOR
Dom-Enum
IN CHORDAL GRAPHS 11
Lemma 16.
Let D be a partial p K , K q -extension of A and suppose that C ˚ p D q exists. Forany feasible p K , K q -extension D in G r V p C ˚ p D qq Y C ˚ p D qs where K “ Q D p C ˚ p D qq , K “pp A Y K q X C ˚ p D qqz K , D Y D is a partial p K , K q -extension of A .Proof. As in the proof of Lemma 11, we choose one private neighbor for vertices in A Y K thathave safe private neighbors in V p C q , C ą C ˚ p D q and let S be the set of these selected vertices.Then we let L : “ t C P L p A Y K q | C ą C ˚ p D q , | C X S | “ u and L : “ t C P L p A Y K q | C ą C ˚ p D q , | C X S | “ u . Let z P S . Now let D ˚ : “ p A Y K Y D Y D q Y ¨˝ ď C P L ,C X S “t y u D C p y q ˛‚ Y ˜ ď C P L D C p z q ¸ . According to the proof of Lemma 11, D ˚ is a feasible p K , K q -extension of A . (cid:3) We can now describe the algorithm.
Algorithm
EnumCombination p G, T , K , K , A, D q G :graph, T :clique tree, K , K : disjoint subsets of C r root of T , A : p K , K q -extendable antichain D : a partial p K , K q -extension of A if C ˚ p D q does not exist then output D ; return K “ Q D p C ˚ p D qq , K : “ pp A Y K q X C ˚ p D qqz K for each D output by EnumKExtension p G r V p C ˚ p D qq Y C ˚ p D qs , T p C ˚ p D qq , K , K q call EnumCombination p G, T , K , K , A, D Y D q end for Lemma 17.
The call
EnumCombination p G, T , K , K , A, A Y K q enumerates all feasible p K , K q -extensions whose top antichain is A in polynomial delay and uses polynomial space.Proof. From Lemma 15, the iteration inputting a partial p K , K q -extension D of A is gener-ated only from the iteration inputting D zp V p C ˚ p D qz C ˚ p D qqq . This assures that the algorithmenumerates all partial p K , K q -extensions of A without duplication. From Lemma 16, there isat least one feasible p K , K q -extension D of A including the partial p K , K q -extension D of A that is the input of the iteration. Thus, all the leaf iterations of the recursion of this algorithmalways outputs a feasible p K , K q -extension of A . Now the delay is bounded by the maximumcomputation time of an iteration multiplied by the depth of the recursion. The depth is at most | V G | , thus the algorithm is polynomial delay since EnumKExtension runs with polynomial delay.Since the depth is at most | V G | , the algorithm is obviously polynomial space. (cid:3) We are now ready to summarize the proof of our main theorem.
Proof of Theorem 1.
By definition every minimal dominating set of G is a feasible pH , Hq -extension. Therefore, the call EnumKExtension p G, T , H , Hq enumerates all minimal dominatingsets in polynomial delay and uses polynomial space by Theorem 13. (cid:3) Conclusion
We have proved that one can list all the minimal dominating sets of a chordal graph with poly-nomial delay and polynomial space. We know from [16] that there exists an output-polynomialalgorithm for the listing of minimal dominating sets of any chordal bipartite graph. It is knownthat chordal bipartite graphs admit a tree-structure similar to the clique tree of chordal graphs.Can we adapt our technique to obtain a polynomial delay and polynomial space algorithm forenumerating the minimal dominating sets of any chordal bipartite graph?Besides the fact that knowing whether there exists an output-polynomial algorithm for listingthe set of minimal dominating sets in a graph is still open, there are some graph classes wherea search for an output-polynomial algorithm deserves to be explored and seems more tractablethan the general problem: we can cite bipartite graphs and unit-disk graphs.
By the reduction in [20] we know that the enumeration of minimal dominating sets in co-bipartite graphs is as hard as the enumeration of minimal dominating sets in all graphs. Can wefind a parameter in co-bipartite graphs that whenever bounded by a function on n (say log p n q )would summarize the tractability of the enumeration of minimal dominating sets in many graphclasses?A related question that arises from the exact algorithm community is the existence of a tightbound of the number of minimal dominating sets in a graph. From [12] we know that the numberof minimal dominating sets in an n -vertex graph is bounded by O p . n q and the best knownlower bound is n { . For several graph classes, including some subclasses of chordal graphs,tight bounds were obtained [7, 17]. Finding a tight upper bound for chordal graphs is still open. References [1] Rakesh Agrawal, Heikki Mannila, Ramakrishnan Srikant, Hannu Toivonen, and A. Inkeri Verkamo. Fastdiscovery of association rules. In
Advances in Knowledge Discovery and Data Mining , pages 307–328.AAAI/MIT Press, 1996.[2] David Avis and Komei Fukuda. Reverse search for enumeration.
Discrete Applied Mathematics , 65(1-3):21–46, 1996.[3] Endre Boros, Khaled M. Elbassioni, and Vladimir Gurvich. Transversal hypergraphs to perfect matchingsin bipartite graphs: Characterization and generation algorithms.
Journal of Graph Theory , 53(3):209–232,2006.[4] Endre Boros, Vladimir Gurvich, Leonid Khachiyan, and Kazuhisa Makino. Dual-bounded generating prob-lems: Partial and multiple transversals of a hypergraph.
SIAM J. Comput. , 30(6):2036–2050, 2000.[5] Endre Boros, Vladimir Gurvich, Leonid Khachiyan, and Kazuhisa Makino. Generating weighted transversalsof a hypergraph. In
RUTGERS UNIVERSITY , pages 13–22, 2000.[6] Bruno Courcelle. Linear delay enumeration and monadic second-order logic.
Discrete Applied Mathematics ,157(12):2675–2700, 2009.[7] Jean-François Couturier, Pinar Heggernes, Pim van ’t Hof, and Dieter Kratsch. Minimal dominating sets ingraph classes: Combinatorial bounds and enumeration.
Theor. Comput. Sci. , 487:82–94, 2013.[8] Reinhard Diestel.
Graph Theory (Graduate Texts in Mathematics) . Springer, 2005.[9] G.A. Dirac. On rigid circuit graphs.
Abhandlungen Aus Dem Mathematischen Seminare der UniversitätHamburg , 25(1-2):71–76, 1961.[10] Thomas Eiter and Georg Gottlob. Identifying the minimal transversals of a hypergraph and related problems.
SIAM J. Comput. , 24(6):1278–1304, 1995.[11] Thomas Eiter, Georg Gottlob, and Kazuhisa Makino. New results on monotone dualization and generatinghypergraph transversals.
SIAM J. Comput. , 32(2):514–537, 2003.[12] Fedor V. Fomin, Fabrizio Grandoni, Artem V. Pyatkin, and Alexey A. Stepanov. Combinatorial bounds viameasure and conquer: Bounding minimal dominating sets and applications.
ACM Transactions on Algo-rithms , 5(1), 2008.[13] Fedor V. Fomin, Pinar Heggernes, Dieter Kratsch, Charis Papadopoulos, and Yngve Villanger. Enumeratingminimal subset feedback vertex sets.
Algorithmica , 69(1):216–231, 2014.[14] Philippe Galinier, Michel Habib, and Christophe Paul. Chordal graphs and their clique graphs. In ManfredNagl, editor, WG , volume 1017 of Lecture Notes in Computer Science , pages 358–371. Springer, 1995.[15] Fanica Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs.
Journal of Com-binatorial Theory, Series B , 16(1):47 – 56, 1974.[16] Petr A. Golovach, Pinar Heggernes, Mamadou M. Kanté, Dieter Kratsch, and Yngve Villanger. Enumeratingminimal dominating sets in chordal bipartite graphs. Submitted, 2014.[17] Petr A. Golovach, Pinar Heggernes, Mamadou M. Kanté, Dieter Kratsch, and Yngve Villanger. Minimaldominating sets in interval graphs and trees. Submitted, 2014.[18] V.A. Gurvich. On theory of multistep games. {USSR} Computational Mathematics and MathematicalPhysics , 13(6):143 – 161, 1973.[19] Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. Enumeration of minimaldominating sets and variants. In Olaf Owe, Martin Steffen, and Jan Arne Telle, editors,
FCT , volume 6914of
Lecture Notes in Computer Science , pages 298–309. Springer, 2011.[20] Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. On the enumerationof minimal dominating sets and related notions. Technical report, Clermont-Université, Université BlaisePascal, LIMOS, CNRS, 2012.[21] Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. On the neighbourhoodhelly of some graph classes and applications to the enumeration of minimal dominating sets. In
ISAAC ,pages 289–298, 2012.[22] Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, and Takeaki Uno. On theenumeration and counting of minimal dominating sets in interval and permutation graphs. In Leizhen Cai,
OLY DELAY FOR
Dom-Enum
IN CHORDAL GRAPHS 13
Siu-Wing Cheng, and Tak Wah Lam, editors,
ISAAC , volume 8283 of
Lecture Notes in Computer Science ,pages 339–349. Springer, 2013.[23] Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, and Takeaki Uno. Polyno-mial delay algorithm for listing minimal edge dominating sets in graphs.
CoRR , abs/1404.3501, 2014.[24] Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled M. Elbassioni, and Vladimir Gurvich. Generatingall vertices of a polyhedron is hard.
Discrete & Computational Geometry , 39(1-3):174–190, 2008.[25] Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled M. Elbassioni, Vladimir Gurvich, and KazuhisaMakino. Generating cut conjunctions in graphs and related problems.
Algorithmica , 51(3):239–263, 2008.[26] Leonid Khachiyan, Endre Boros, Khaled M. Elbassioni, and Vladimir Gurvich. On enumerating minimaldicuts and strongly connected subgraphs.
Algorithmica , 50(1):159–172, 2008.[27] Eugene L. Lawler, Jan Karel Lenstra, and A. H. G. Rinnooy Kan. Generating all maximal independent sets:Np-hardness and polynomial-time algorithms.
SIAM J. Comput. , 9(3):558–565, 1980.[28] Arnaud Mary.
Énumération des Dominants Minimaux d’un graphe . PhD thesis, Université Blaise Pascal,2013.[29] K.G. Ramamurthy.
Coherent Structures and Simple Games . Theory and decision library, Game theory,mathematical programming, and operations research: Series C. Springer, 1990.[30] Benno Schwikowski and Ewald Speckenmeyer. On enumerating all minimal solutions of feedback problems.
Discrete Applied Mathematics , 117(1-3):253–265, 2002.[31] Yann Strozecki.
Enumeration Complexity and Matroid Decomposition . PhD thesis, Université Paris Diderot- Paris 7, 2010.[32] Robert Endre Tarjan. Enumeration of the elementary circuits of a directed graph.
SIAM J. Comput. ,2(3):211–216, 1973.[33] J.D. Ullman.
Principles of Database and Knowledge-Base Systems . Number vol. 1 in Principles of ComputerScience Series. Computer Science Press, 1989.
Clermont-Université, Université Blaise Pascal, LIMOS, CNRS, France
E-mail address : {mamadou.kante,limouzy,mary,nourine}@isima.fr National Institute of Informatics, Japan
E-mail address ::