On the maximum number of minimal connected dominating sets in convex bipartite graphs
aa r X i v : . [ c s . D S ] A ug On the maximum number of minimal connecteddominating sets in convex bipartite graphs ∗ Mohamed Yosri Sayadi
Université de Lorraine, LGIPM, F-57000 Metz, [email protected]
Abstract
The enumeration of minimal connected dominating sets is known to be notoriously hard forgeneral graphs. Currently, it is only known that the sets can be enumerated slightly fasterthan O ∗ (2 n ) and the algorithm is highly nontrivial. Moreover, it seems that it is hard to usebipartiteness as a structural aide when constructing enumeration algorithms. Hence, to the best ofour knowledge, there is no known input-sensitive algorithm for enumerating minimal dominatingsets, or one of their related sets, in bipartite graphs better than that of general graphs. In thispaper, we provide the first input-sensitive enumeration algorithm for some non trivial subclass ofbipartite graphs, namely the convex graphs. We present an algorithm to enumerate all minimalconnected dominating sets of convex bipartite graphs in time O (1 . n ) where n is the numberof vertices of the input graph. Our algorithm implies a corresponding upper bound for thenumber of minimal connected dominating sets for this graph class. We complement the result byproviding a convex bipartite graph, which have at least 3 ( n − / minimal connected dominatingsets. G.2.2, F.2.2
Keywords and phrases
Minimal connected dominating set, exact algorithms, enumeration, graphs
Listing, generating or enumerating objects of specified type and properties has importantapplications in various domains of computer science, such as data mining, machine learning,and artificial intelligence, as well as in other sciences, especially biology. In particular,enumeration algorithms whose running time is measured in the size of the input have gainedincreasing interest recently [17, 18, 11].In fact, several classical examples exist in this direction, of which one of the most fam-ous is perhaps that of Moon and Moser [13] who showed that the maximum number ofmaximal independent sets in a graph on n vertices is Θ(3 n/ ). Recently Lokshtanov et al.[1] studied the maximum number of minimal connected dominating sets in an arbitrary n -vertex graph and they showed that the maximum number of minimal connected dominatingsets is O (2 (1 − ǫ ) n ) where ǫ > − . Contrary to the number of maximal independent setswhere the upper and lower bounds are tight, the best known lower bound for the numberof minimal connected dominating sets is 3 ( n − / [2], meaning there is a huge gap up tonow between the lower and upper bounds. It is worth noting that computing a minimumconnected dominating set is one of the classical NP-hard problems as already mentioned inthe monograph of Garey and Johnson [14]. Furthermore, it also is NP-hard for bipartitegraphs [7] and chordal bipartite graphs [8]. The best known running time of an algorithmsolving this problem for general graphs is O (1 . n ) [3]. However, the minimum connected ∗ This work is supported by the French National Research Agency by the ANR project GraphEn (ANR-15-CE40-0009). :2 On the maximum number of minimal connected dominating sets in convex bipartite graphs dominating set problem is tractable for convex bipartite graphs and it can be solved in O ( n )time [5]. Furthermore, the problems of finding a minimum dominating set and a minimumindependent dominating set in an n -vertex convex bipartite graph are solvable in time O ( n )[6]. Despite the structural and algorithmic properties of bipartite graphs, there is no enu-meration algorithm for the minimal connected dominating sets in a bipartite graph betterthan that for general graphs. The situation is similar when considering the enumeration ofminimal dominating sets and maximal irredundant sets in bipartite graphs. In this paperwe study the enumeration and maximum number of minimal connected dominating sets inconvex bipartite graphs, and we prove that the number of minimal connected dominatingsets in a convex bipartite graph is O (1 . n ) and that those sets can be enumerated intime O (1 . n ).The studied graph classes for the enumeration of minimal connected dominating sets aresummarized in the following table, where n is the number of vertices and m is the numberof edges of an input graph belonging to the given class. Table 1
Lower and upper bounds on the maximum number of minimal connected dominatingsets.
Graph class Lower bound Ref. Upper bound Ref. general 3 ( n − / [2] O (2 (1 − ǫ ) n ) [1]chordal 3 ( n − / [2] O (1 . n ) [2]split 1 . n [10] 1 . n [4]co-bipartite 1 . n [10] n + 2 × . n [4]interval 3 ( n − / [2] 3 ( n − / [2]AT-free 3 ( n − / [2] O ∗ (3 ( n − / ) [2]strongly chordal 3 ( n − / [2] 3 n/ [2]distance-hereditary 3 ( n − / [2] 3 n/ × n [2]cograph m [2] m [2]convex bipartite 3 ( n − / [this paper] O (1 . n ) [this paper] We consider finite undirected convex bipartite graphs G = ( U, W, E ) without loops or mul-tiple edges. Let V = U ∪ W be the vertex set of G , where U and W define the bipartition ofvertices. We also let n = | V ( G ) | and m = | E ( G ) | denote the number of vertices and edges,respectively, of the input graph G . A bipartite graph G = ( U, W, E ) is convex if there existsan ordering of the vertices of W such that for each u ∈ U , the neighbors of u are consecutivein W . For convenience, we consider that U = { , , . . . , | U |} and W = { w , w , . . . , w | W | } ,and that the vertices in W are given according to the ordering mentioned above. We saythat a vertex w i ∈ W is smaller (larger) than a vertex w j ∈ W if the integer i is smaller(larger) than the integer j . By the definition of convex bipartite graphs, the neighbors ofa vertex u ∈ U can be represented as an interval I u = [ l ( I u ) , r ( I u )] , called the neighborinterval of u , where l ( I u ) and r ( I u ) are the smallest and largest vertices, respectively, in theinterval of vertices of W adjacent to u . Further, we call l ( I u ) and r ( I u ) the left endpointand right endpoint of the interval I u , respectively. Then, the neighbors of vertices of U can be represented by a set I ( U ) of intervals. We call I ( U ) the interval representation of . Y. Sayadi 1:3 neighbors of vertices in U . A star graph is the complete bipartite graph K ,n − : a tree withone internal node and n − v be a vertex of V and let D be a subset of V . The open neighborhood N ( v ) ofthe vertex v consists of the set of vertices adjacent to v , that is, N ( v ) = { w ∈ V | vw ∈ E } ,and the closed neighborhood of v is N [ v ] = N ( v ) ∪ { v } . The open neighborhood N ( D )is defined to be ∪ v ∈ D N ( v ), and the closed neighborhood of D is N [ D ] = N ( D ) ∪ D . Wedenote by deg ( v ) = | N ( v ) | the degree of v . The set D dominates vertex v if either v ∈ D or N ( v ) ∩ D = ∅ . If D dominates all vertices in a subset S of V , then we say that D dominates S . The set D is called a dominating set of G if and only if D dominates V . A vertex u ∈ V is a private neighbor of the vertex v (with respect to D ) if u is dominated by v but is notdominated by D \{ v } . We denote by G [ D ] the subgraph of G induced by D . A set D ⊆ V ( G )is connected if G [ D ] is a connected graph. A set of vertices D is a connected dominating setif D is connected and D dominates V ( G ). A (connected) dominating set is minimal if noproper subset of it is a (connected) dominating set. Further, a connected dominating set D is minimal if and only if for any v ∈ D , v has a private neighbor or D \ { v } is disconnected,i.e., v is a cut vertex of G [ D ]. Notice that every cut vertex of G belongs to every minimalconnected dominating set of G .It is possible to recognize convex bipartite graphs in linear time[15]. We use the O ∗ notation that hides polynomials, i.e., we write f ( n ) = O ∗ ( g ( n )) if f ( n ) = O ( p ( n ) · g ( n ))where p is a polynomial in n . In this section, we will provide some useful properties which any minimal connected domin-ating set D of a convex bipartite graph G = ( U, W, E ) satisfies. Recall that the vertices W of the graph G = ( U, W, E ) are given with an ordering w , w , . . . , w | W | satisfying that forall u ∈ U , N ( u ) ⊂ W is an interval I u = [ l ( u ) , r ( u )]. ◮ Observation 1.
A set D ⊆ V is a minimal connected dominating set of a K ,n − if andonly if D is a singleton consisting of the internal node of K ,n − . Hence, if G is a star graphthen the problem can be solved in linear time.From now on, we consider a convex bipartite graph G = ( U, W, E ) satisfying | W | ≥ | U | ≥ | D | ≥ G . ◮ Observation 2.
Each vertex v ∈ V has a neighbor x ∈ D . Proof. If v D , then there exists an x ∈ D ∩ N ( v ) because D is a dominating set. Otherwise v ∈ D , and then since D is a connected set and | D | ≥
2, there exists an x ∈ D ∩ N ( v ). ◭◮ Lemma 1. If i, j ∈ U such that I i ⊆ I j then |{ i, j } ∩ D | ≤ . Proof.
Let i, j ∈ U : I i ⊆ I j and suppose, for the sake of contradiction, that both i, j ∈ D .The vertex i cannot have a private neighbor with respect to D in W because all its neighborsare dominated by j . Furthermore, i cannot be a private for itself because D is connectedand | D | ≥
2. Therefore i cannot have a private neighbor. On the other hand, to show that i is not a cut vertex of G [ D ] let x, y ∈ V \ { i } . For every x − y path in G which passesthrough i , there exists a x − y path passing through j without passing through i . Therefore i cannot be a cut vertex of G [ D ]. Hence D is not a minimal connected dominating set; acontradiction. ◭ This immediately implies :4 On the maximum number of minimal connected dominating sets in convex bipartite graphs ◮ Lemma 2. If r ( I i ) = r ( I j ) or l ( I i ) = l ( I j ) , i, j ∈ U , then |{ i, j } ∩ D | ≤ . The following two lemmata are crucial for our branching algorithm. ◮ Lemma 3.
For every i ≤ | W | − , there exists a vertex u ∈ D ∩ U such that u ∈ N ( w i ) ∩ N ( w i +1 ) . Proof.
Suppose, for the sake of contradiction, there exists an i ≤ | W | − D ∩ N ( w i ) ∩ N ( w i +1 ) = ∅ . Let us consider the induced subgraph H = G [ W ∪ ( U ∩ D )]. We claimthat H ′ = ( U ′ , W ′ , E ′ ) and H ′′ = ( U ′′ , W ′′ , E ′′ ) such that U ′ = { u ′ ∈ U : r ( I u ′ ) ≤ w i } , W ′ = { w i ′ ∈ W : w i ′ ≤ w i } , U ′′ = { u ′′ ∈ U : l ( I u ′′ ) ≥ w i +1 } and W ′′ = { w i ′′ ∈ W : w i ′′ ≥ w i } ,are two connected components of H = ( U, W, E ). To see this note that there is no w j u ∈ E such that w j ∈ W ′ , u ∈ U ′′ or w j ∈ W ′′ , u ∈ U ′ , otherwise u ∈ N ( w i ) ∩ N ( w i +1 ). Hence theinduced subgraphs H ′ and H ′′ are indeed two disjoint components of H . By observation 2, N ( w i ) ∩ D = ∅ and N ( w i +1 ) ∩ D = ∅ , thus D ∩ U ′ = ∅ and D ∩ U ′′ = ∅ . Hence G [ D ] isdisconnected, a contradiction. ◭ Similarly, we may obtain the following lemma. ◮ Lemma 4.
For any two consecutive w i , w j ∈ W ∩ D in G [ D ] (not necessarily consecutivein W ), there exists a vertex u ∈ D ∩ U such that u ∈ N ( w i ) ∩ N ( w j ) . ◮ Lemma 5.
Let i, j ∈ U such that I i ∩ I j = ∅ . If there is a k ∈ U such that I k ⊂ ( I i ∪ I j ) and i, j, k ∈ D ∩ U , then I i ∩ I j ∩ D = ∅ . Proof.
Let i, j, k ∈ D ∩ U , I i ∩ I j = ∅ and I k ⊂ { I i ∪ I j } . Suppose, for the sake ofcontradiction, that there is a w ∈ D ∩ I i ∩ I j . It is clear that k is a cut vertex of G [ D ].Therefore there are w x , w ′ x ∈ D ∩ I k and w x , w ′ x are in two different components of G [ D \ k ].Furthermore w x ∈ I i ∩ I k \ I j and w ′ x ∈ I j ∩ I k \ I i . However there is a w x − w ′ x path passingthrough i, w, j without passing through k , a contradiction. ◭◮ Lemma 6.
Let i, j ∈ D ∩ U such that I i ∩ I j = ∅ . Then |{ k ∈ D : I k ⊂ ( I i ∪ I j ) }| ≤ . Proof.
Let i, j ∈ D , I i ∩ I j = ∅ and suppose, for the sake of contradiction, that there are k, l ∈ D satisfying I k ⊂ { I i ∪ I j } and I l ⊂ { I i ∪ I j } . Because of lemmata 2 and 1 andw.l.o.g, then l ( I i ) < l ( I k ) < l ( I l ) < l ( I j ) and r ( I i ) < r ( I k ) < r ( I l ) < r ( I j ). It is clear that k, l are cut vertices of G [ D ]. As k is a cut vertex of G [ D ], there are w x , w x ′ ∈ D ∩ I k and w x , w ′ x in two different components of G [ D \ k ]. Furthermore, at least one of them doesnot belong to I l . W.l.o.g, let w x < w x ′ and w x / ∈ I l . Since l is a cut vertex of G [ D ], thereare w y , w y ′ ∈ D ∩ I l and w y , w y ′ are in two different components of G [ D \ l ]. It is clearthat at least one of w y , w y ′ is not a neighbor of k . W.l.o.g, let w y ′ > w y and w y ′ > r ( I k ).Furthermore, w y < l ( I j ) and w y > w x otherwise w x , w x ′ will be adjacent to l and w y , w y ′ will be adjacent to j . It is clear that w x ′ > r ( I i ). Hence, there is a w y − w y ′ path passingthrough k, w x ′ , j without passing through l , a contradiction. ◭ Now we are ready to present our algorithm.
The basic idea of the enumeration algorithm is to choose the vertices of a minimal connecteddominating set D by using reduction and branching rules. Furthermore the algorithm is par-titioned into stages. During the preprocessing (stage 1) a collection of initializing recursivecalls is done. In stage 2 we choose the vertices of D ∩ U . Therefore when we select a vertex . Y. Sayadi 1:5 u ∈ U , we add it immediately to the minimal connected dominating set D . However whenwe discard u from D , we move it to T in order to dominate it in the next stages of thealgorithm by a (still to be selected) vertex of W . Furthermore when we fix the vertices of D ∩ U , we mark some vertices of W as forbidden vertices F , which means that those verticesof W cannot be selected (they are excluded from D). In stage 3 we mainly remove all thosepartial solutions of stage 2 that cannot be extended to a minimal connected dominatingset of G . Finally in stage 4 the remaining partial solutions are completed into minimalconnected dominating set, if possible. Stage 1. Preprocessing.
We consider the following procedure
EnumLevel1 ( U, W ), where we initialize the call of
EnumLevel2 ( u, U, D, T, F ). Step 1. for each u ∈ N ( w ), call EnumLevel2 ( u, U \ N ( w ) , { u } , N ( w ) \ { u } , ∅ ).By observation 2, | N ( w ) ∩ D | ≥
1. Since all the neighbors of w have the same left endpoint, | N ( w ) ∩ D | ≤ | N ( w ) ∩ D | = 1. Hence we add exactly one vertex u ∈ N ( w ) to D and discard the other vertices N ( w ) \ { u } from the solution, i.e. we aremoving them to T in order to dominate them by vertices of W in the next stages of thealgorithm. Finally, we initialize F = ∅ and call EnumLevel2 ( u, U, D, T, F ). Stage 2.
We consider the following recursive procedure
EnumLevel2 ( u, U, D, T, F ), where u is thevertex already selected in D ∩ U with the largest right endpoint and U, D, T, F were men-tioned above. Let r ( I u ) be r . Step 1.
If there is a vertex i ∈ U such that r ( I i ) ≤ r , then call EnumLevel2 ( u, U \{ i } , D, T ∪ { i } , F ). Step 2. If r = w | W | , then call EnumLevel3 ( D, T, F ). Step 3. If N ( r ) = ∅ , then stop. Step 4. If deg ( r ) = 1, then let { j } = N ( r ) and call EnumLevel2 ( j, U \ { j } , D ∪ { j } , T, F ). Step 5. If deg ( r ) = 2, then let { j, k } = N ( r ) and branch as follows:If I j ⊆ I k or I k ⊆ I j , then branch:(i) call EnumLevel2 ( j, U \ { j, k } , D ∪ { j } , T ∪ { k } , F ),(ii) call EnumLevel2 ( k, U \ { j, k } , D ∪ { k } , T ∪ { j } , F ).Else, let r ( j ) > r ( k ) and branch:(i) call EnumLevel2 ( j, U \ { j, k } , D ∪ { j } , T ∪ { k } , F ),(ii) call EnumLevel2 ( k, U \ { j, k } , D ∪ { k } , T ∪ { j } , F ).(iii) call EnumLevel2 ( j, U \ { j, k } , D ∪ { j, k } , T, F ∪ ( I j ∩ I u )). Step 6. If deg ( r ) ≥
3, then let j be the neighbor of r with the largest right endpoint andbranch as follows:(i) call EnumLevel2 ( j, U \ N ( r ) , D ∪ { j } , T ∪ N ( r ) \ { j } , F ),(ii) for each x ∈ N ( r ) with I x I j , call EnumLevel2 ( j, U \ N ( r ) , D ∪ { x, j } , T ∪ N ( r ) \ { x, j } , F ∪ { I j ∩ I u } ).(iii) call EnumLevel2 ( u, U \ { j } , D, T ∪ { j } , F ). :6 On the maximum number of minimal connected dominating sets in convex bipartite graphs Let j be the neighbor of r with the largest right endpoint. Either j ∈ D or j / ∈ D .If j / ∈ D , then in case (iii) we add j to T in order to be dominated in the next stages byvertices of W . Therefore we call EnumLevel2 ( u, U \ { j } , D, T ∪ { j } , F ).Suppose now that j ∈ D . Because j is the neighbor of r with the largest right endpointthen I u ∩ I j = ∅ and for any x ∈ D ∩ N ( r ) \ { u, j } we have I x ⊂ ( I u ∪ I j ). Thereforeby lemma 6, | N ( r ) ∩ D \ { u, j }| ≤
1. If N ( r ) ∩ D \ { u, j } = ∅ , then in case (i) we call EnumLevel2 ( j, U \ N ( r ) , D ∪ { j } , T ∪ N ( r ) \ { j } , F ). Suppose now that N ( r ) ∩ D \ { u, j } = { x } . In this case, we forbid by lemma 5 ( I j ∩ I u ). Notice that by lemma 1 I x I j . Hencewe branch for each x ∈ N ( r ) satisfying I x I j and we call EnumLevel2 ( j, U \ N ( r ) , D ∪ { x, j } , T ∪ N ( r ) \ { x, j } , F ∪ { I j ∩ I u } ). If there is an x ∈ U such that I x ⊆ I j , then x will be treated by step 1 in the recursive call. Stage 3.
In this stage, the procedure
EnumLevel3 ( D, T, F ) deletes the bad partial solutions, i.e.those that can definitely not be extended into a minimal connected dominating set of G ,generated in the previous stage and it preprocesses the remaining partial solutions for thenext stage. Let J ( T, D ) be a set of intervals, where for every interval we have to select atleast one corresponding vertex either to dominate a vertex in T or to connect vertices in D . Hence we initialize J ( T, D ) by the interval representations of neighbors of vertices in T and consider the induced graph G [ W ∪ ( U ∩ D )]. Let I ′ be the interval representations ofneighbors of vertices in D ∩ U . Step 1.
If there exist u, v ∈ D ∩ U such that I ′ u ⊆ I ′ v , then stop. Step 2.
While ( I ′ = ∅ )BeginLet I ′ i be the interval with the smallest right endpoint in I ′ and r ( I ′ i ) be r .If I ′ = { I ′ i } , then call EnumLevel4 ( D, J, F ).Else if deg ( r ) >
3, then stop.Else if deg ( r ) = 2, then let N ( r ) = { i, j } : J ( T, D ) ← J ( T, D ) ∪ { I ′ i ∩ I ′ j } ; I ′ ← I ′ \ I ′ i .Else if deg ( r ) = 3, then let N ( r ) = { i, j, k } , I ′ j be the interval with the largest rightendpoint and I ′ k be the other interval such as I ′ k ⊂ ( I ′ i ∪ I ′ j ) : J ( T, D ) ← J ( T, D ) ∪ ( I ′ i ∩ I ′ k \ I ′ j ) ∪ ( I ′ j ∩ I ′ k \ I ′ i ) ; I ′ ← I ′ \ ( I ′ i ∪ I ′ k ).End.First, every vertex in T should be dominated by at least one of its neighbors. Therefore weinitialized J ( T, D ) by the interval representations of neighbors of vertices in T .Now if deg ( r ) > G [ W ∪ ( U ∩ D )], then D cannot be a minimalconnected dominating set by lemma 6. Therefore we stop.So let deg ( r ) = 2:Suppose, for the sake of contradiction, that I ′ i ∩ I ′ j ∩ D = ∅ . Then by observation 2 there isa w i ∈ I ′ i ∩ D and there is a w j ∈ I ′ j ∩ D . W.l.o.g, let w i , w j be consecutive in G [ D ]. As deg ( r ) = 2, there cannot be any u ∈ U ∩ D such that u ∈ N ( w i ) ∩ N ( w j ), a contradictionby lemma 4. Therefore I ′ i ∩ I ′ j ∩ D = ∅ . Hence J ( T, D ) ← J ( T, D ) ∪ { I ′ i ∩ I ′ j } .Now if deg ( r ) = 3, then the vertices of I ′ i ∩ I ′ j are forbidden and k is a cut vertex of G [ D ] whichconnects two vertices in W ∩ D . If both of these two vertices had belonged to ( I ′ i ∩ I ′ k \ I ′ j ) or( I ′ j ∩ I ′ k \ I ′ i ), then k would not have been a cut vertex. Therefore one belongs to ( I ′ i ∩ I ′ k \ I ′ j ) . Y. Sayadi 1:7 and the another one belongs to ( I ′ j ∩ I ′ k \ I ′ i ). Hence, we added ( I ′ i ∩ I ′ k \ I ′ j ) and ( I ′ j ∩ I ′ k \ I ′ i )to J ( T, D ). Stage 4.
We consider the following recursive procedure called
EnumLevel4 ( W, J, D ). We try, in thislevel, to select at least one vertex of each interval in J in order to dominate the vertices of T and to connect D by some vertices of W .Let J i ∈ J be the interval with the smallest right endpoint. If we have more than onecandidate interval, then the shortest interval amongst them will be chosen. Step 1. If J = ∅ , then check whether D is a minimal connected dominating set of G andoutput it if it holds; then stop. Step 2.
If all the vertices of J i are forbidden, then stop. Step 3.
For each non forbidden w ∈ J i , call EnumLevel4 ( W \ J i , J \ { J k : w ∈ J k } , D ∪ { w } ).By the construction of J , we must select at least one vertex of each interval in J . Therefore | D ∩ J i | ≥
1. Hence in Step 2, we stop when all the vertices of J i are forbidden. Let us provenow that | D ∩ J i | ≤
1. Suppose, for the sake of contradiction, that | D ∩ J i | ≥
2. Thereforethere are w i , w j ∈ J i ∩ D . W.l.o.g, we suppose that w i < w j . In this case if w i ∈ J k such that J k ∈ J , then w j ∈ J k . Therefore, D is not minimal because we can delete w i and D is stilla connected dominating set, a contradiction. Hence | D ∩ J i | ≤ | D ∩ J i | ≥ | D ∩ J i | = 1. We establish an upper bound on the number of minimal connected dominating sets in convexbipartite graphs, via the branching algorithm described in the previous section and itsrunning-time analysis.For the analysis of the running time and the number of minimal connected dominatingsets that are produced by such an algorithm, we use a technique based on solving recurrencesfor branching steps and branching rules respectively. We refer to the book by Fomin andKratsch [16] for a detailed introduction. To analyze such a branching algorithm solving anenumeration problem, one assigns to each instance I of the recursive algorithm a measure µ ( I ) that one may consider as the size of the instance I . If the algorithm branches on aninstance I into t new instances, such that the measure decreases by c , c , . . . , c t for eachnew instance, respectively, we say that ( c , c , . . . , c t ) is the branching vector of this step.We find the unique positive real root α , called a branching number , of the characteristicpolynomial p ( x ) = x c − x c − c − . . . − x c − c t for c = max { c , . . . , c t } . Then standard analysis(see [16]), shows that if µ ( I ) ≤ n for all instances I , the number of leaves of the search treeproduced by an execution of the algorithm is O ∗ ( α n ), where α is the maximum value of thebranching numbers over all branching vectors that occur in the algorithm. This approachallows us to achieve running times of the form O ∗ ( α n ) for some real α ≥
1. As the numberof minimal connected dominating sets produced by an algorithm is upper bounded by thenumber of leaves of the search tree, we also obtain the upper bound for the number ofminimal connected dominating sets of the same form O ∗ ( α n ). If α has been obtained byrounding up then one may replace O ∗ ( α n ) by O ( α n ); see [16].To analyze the running time of the algorithm, we compute the branching vectors for allbranching steps of the procedures EnumLevel4 ( W, J, D ) and
EnumLevel2 ( u, U, D, T, F ). :8 On the maximum number of minimal connected dominating sets in convex bipartite graphs Notice that
EnumLevel1 and
EnumLevel3 runs in polynomial time as no branching isneeded. We set the measure of an instance to | U | + | W \ F | . Hence by moving a vertex u ∈ U to T or by forbidding a vertex w ∈ W , the measure of the instance decreases by 1.Let us start with EnumLevel2 ( u, U, D, T, F ). Notice that in Steps 1 − Step 5.
The first branching vector in Step 5 is (2 , W . Thus the branching vector is (2 , , , ,
3) and thus α < . Step 6.
It is straightforward to see, that the maximum value of the branching number isachieved if | I u ∩ I j | = 1 and if there is no I x ⊆ I j such that x, j ∈ N ( r ). Thus we branch forall the neighbors of r and we forbid in each case the only vertex I u ∩ I j . The correspondingbranching vector is ( t, t + 1 , . . . , t + 1 | {z } t − ,
1) where t = deg ( r ) ≥
3, and the maximum value ofthe branching number α < . t = 3.For EnumLevel4 ( W, J, D ), we need to analyze Step 3 only because Steps 1 and 2 arereduction rules. The corresponding branching vector is ( t, . . . , t | {z } t ) where t = | J i \ F | , and themaximum value of the branching number α < . t = 3.The largest branching number is (majorized by) 1 . ◮ Theorem 7.
A convex bipartite graph has at most O (1 . n ) minimal connected domin-ating sets, and these can be enumerated in time O (1 . n ) . To obtain a lower bound for the maximum number of minimal connected dominating setsin a convex bipartite graph, we use a slight modification of the lower bound obtained in [2]. ◮ Proposition 1.
There are convex bipartite graphs with at least 3 ( n − / minimal con-nected dominating sets. Proof.
To obtain the bound for convex bipartite graphs, consider the graph G constructedas follows for a positive odd integer k .For i ∈ { , . . . , k } , construct a triple of independent vertices T i = { x i , y i , z i } .For i ∈ { , . . . , k } , join each vertex of T i − with every vertex of T i by an edge.Construct two vertices u and v and edges ux , uy , uz and vx k , vy k , vz k .Clearly, G has n = 3 k + 2 vertices. Notice that D ⊆ V ( G ) is a minimal connecteddominating set of G if and only if u, v / ∈ D and | D ∩ T i | = 1 for i ∈ , . . . , k . Therefore, G has 3 k = 3 ( n − / minimal connected dominating sets. It remains to observe that G isconvex bipartite from its following model. ◭ . Y. Sayadi 1:9 x • y • z • x • y • z • x • y • z • x k − • y k − • z k − • x k • y k • z k • I u I x I y I z I x I y I z I v I x k − I y k − I z k − While the enumeration of the minimal dominating sets of a graph attracted a lot of atten-tion, in particular due to its relation to the (output-sensitive) enumeration of the minimaltransversals of a hypergraph, not much is known for the highly related problem of enumer-ation of the minimal connected dominating sets for a variety of graph classes and for bothinput- and output-sensitive enumeration. First (input-sensitive) enumeration algorithms forminimal connected dominating set of a graph have been obtained in [2] which studies amongothers strongly chordal and AT-free graphs. It seems that it is hard to use bipartiteness asa structural aide when constructing enumeration algorithms. In this paper we provided thefirst enumeration algorithm for some non trivial subclass of bipartite graphs, namely the con-vex graphs. It would be very interesting to extend our research to enumeration algorithmsand upper bounds on the number of minimal connected dominating set in chordal bipart-ite graphs and bipartite graphs, on the number of minimal dominating sets and maximalirredundant sets in convex bipartite graphs on the other hand.
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