Edge Degeneracy: Algorithmic and Structural Results
Stratis Limnios, Christophe Paul, Joanny Perret, Dimitrios M. Thilikos
EEdge Degeneracy: Algorithmic and Structural Results
Stratis Limnios ∗† Christophe Paul ‡†§
Joanny Perret ¶ Dimitrios M. Thilikos ‡†§k
Abstract
We consider a cops and robber game where the cops are blocking edges of a graph, while therobber occupies its vertices. At each round of the game, the cops choose some set of edgesto block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges ( s can be seen as the speed of the robber). Both parts have complete knowledgeof the opponent’s moves and the cops win when they occupy all edges incident to the robbersposition. We introduce the capture cost on G against a robber of speed s . This defines ahierarchy of invariants, namely δ , δ , . . . , δ ∞ e , where δ ∞ e is an edge-analogue of the admissibilitygraph invariant, namely the edge-admissibility of a graph. We prove that the problem askingwether δ s e ( G ) ≤ k , is polynomially solvable when s ∈ { , , ∞} while, otherwise, it is NP -complete. Our main result is a structural theorem for graphs of bounded edge-admissibility.We prove that every graph of edge-admissibility at most k can be constructed using ( ≤ k )-edge-sums, starting from graphs whose all vertices, except possibly from one, have degree atmost k . Our structural result is approximately tight in the sense that graphs generated by thisconstruction always have edge-admissibility at most 2 k −
1. Our proofs are based on a precisestructural characterization of the graphs that do not contain θ r as an immersion, where θ r isthe graph on two vertices and r parallel edges. Keywords:
Graph Admissibility, Graph degeneracy, Graph Searching, Cops and robber games,Graph decomposition theorems.
All graphs in this paper are undirected, finite, loopless, and may have parallel edges. We denoteby V ( G ) the set of vertices of a graph G , while we denote by E ( G ) the multi-set of its edges. Wealso use the term s -path of G for a path of G that has length at most s .A ( k, s ) -hide out in a graph G is a subset S of its vertices such that, for each vertex v ∈ S , itis not possible to block all s -paths from v to the rest of S by less than k vertices, different than v . The s -degeneracy of a graph G , has been introduced in [23] as the minimum k for which G contains a ( k, s )-hide out. s -degeneracy defines a hierarchy of graph invariants that, when s = 1, ∗ Data Science and Mining (DaSciM) team, Laboratoire d’Informatique (LIX) ´Ecole Polytechnique, Palaiseau,France. † Supported by project ESIGMA (ANR-17-CE23-0010). ‡ LIRMM, Univ Montpellier, CNRS, Montpellier, France. § Supported by project DEMOGRAPH (ANR-16-CE40-0028). ¶ GIPSA-lab, Universit´e de Grenoble, Grenoble, France. k Supported by the Research Council of Norway and the French Ministry of Europe and Foreign Affairs, via theFranco-Norwegian project PHC AURORA 2019. a r X i v : . [ c s . D M ] S e p ives the classic invariant of graph degeneracy [5, 18, 20] and, when s = ∞ , gives the parameter of ∞ -admissibility that was introduced by Dvoˇr´ak in [10] and studied in [6, 9, 15, 17, 21, 22, 24].In this paper we introduce and study the edge analogue of the above hierarchy of graph in-variants, namely the s -edge-degeneracy hierarchy . The new parameter results from the one of s -degeneracy if we replace ( k, s )-hide outs by ( k, s ) -edge hide outs where we ask that, for eachvertex v of S , it is not possible to block all s -paths from v to the rest of S by less than k edges. Itfollows that the value of s -edge-degeneracy may vary considerably than the one of s -degeneracy. Forinstance, consider the graph θ k consisting of two vertices and k parallel edges between them. It iseasy to see that, for every positive integer s , the s -degeneracy of θ k is 2, while it s -edge-degeneracyis k (the two vertices form a ( k, s )-edge hideout). In other words, s -edge-degeneracy can be seenas an alternative way to extent the notion of degeneracy using edge separators instead of vertexseparators.In Subsection 3.1 we introduce two alternative definitions for s -edge-degeneracy, apart from theone using ( k, s )-edge hide outs. The first is in terms of a graph searching game and the second is interms of graph layouts. Next, we prove a min-max theorem supporting the equivalence of the threedefinitions. As a consequence of this theorem, we can identify the computational complexity of s -edge-degeneracy: it can be computed in polynomial time when s ∈ { , , ∞} , while for all othervalues of s , desiding whether its value is at most k is an NP -complete problem.Our next step is to provide a structural theorem for the ∞ -edge-degeneracy that, from now on,we call ∞ -edge-admissibility. For ∞ -degeneracy (also known as ∞ -admissibility), Dvoˇr´ak provedthe following structural characterization [9, Theorem 6]. Proposition 1.1.
For every k , there exist constants d k , c k and a k such that every graph G with ∞ -admissibility at most k can be constructed by applying ( ≤ c k ) -clique sums starting from graphswhere at most d k vertices have degree at least a k . In the above proposition the ( ≤ k ) -clique sum operation receives as input two graphs G and G such that each G i contains a clique K i with vertex set { v i , . . . , v iρ } , ρ ≤ k . The outcome ofthe operation is the graph occurring if we identify v j and v j for j ∈ { , . . . , ρ } and then removesome of the edges between the identified vertices. While the constants of Proposition 1.1 where notspecified in [9], an alternative proof was recently given by Weißauer in [24] where d k = k , c k = k ,and a k = 2 k ( k − ∞ - edge - admissibility that is thefollowing Theorem 1.2.
For every k , every graph G with ∞ - edge - admissibility at most k can be constructedby applying ( ≤ k ) -edge sums starting from graphs where at most one vertex has degree at least k + 1 . Observe that Theorem 1.2 occurs from Proposition 1.1 if we replace ∞ - admissibility by ∞ - edge - admissibility , if, instead of clique sums, we consider edge sums, and if we set d k = 1, c k = k , and a k = k + 1. The ( ≤ k )-edge sum operation (the definition is postponed to Subsection 4.1) wasdefined in [25] (see also [13]) and can be seen as the edge-counterpart of clique sums.The proof of our structural theorem is derived by a precise structural characterization of thegraphs where each pair of vertices is separated by a cut of size at most k . We prove that thesegraphs are exactly those that can be constructed using ( ≤ k )-edge sums from graphs where all2ut one of their vertices have degree at most k . This directly implies our structural theorem for ∞ - edge - admissibility , as every pair of two vertices linked by k + 1 pairwise edge-disjoint paths is a( k + 1 , ∞ )-edge hide out.Our last result is that the converse of the structural characterization in Theorem 1.2 holds inan approximate way: if G can be constructed using ( ≤ k )-edge sums from graphs where all but oneof their vertices have degree at most k , then the ∞ -edge-admissibility of G is at most 2 k −
1. Thissuggests that our decomposition theorem is indeed the correct choice for the parameter of ∞ -edgeadmissibility. Sets and integers.
Given a non-negative integer s , we denote by N ≥ s the set of all non-negativeintegers that are not smaller than s . We also denote N + ≥ s = N ≥ s ∪ {∞} . Given two integers p ≤ q, we set [ p, q ] = { p, p + 1 , . . . , q } and given a k ∈ N ≥ we define [ k ] = [1 , k ] . Given a set A, we use 2 A for the set of all its subsets, we define (cid:0) A (cid:1) := { S | S ∈ A and | S | = 2 } , and, given a k ∈ N ≥ wedenote by A ( ≤ k ) the set of all subsets of A that have size at most k . A near-partition of a set A is acollection of pairwise disjoint sets whose union is A . A bipartition of A, | A | ≥ A into two non-empty sets. Graphs.
All graphs in this paper are undirected, finite, loopless, and may have parallel edges.We denote by V ( G ) the set of vertices of a graph G while we use E ( G ) for the multi-set of itsedges. Given a graph G and a vertex v , we define E G ( v ) as the multi-set of all edges of G that areincident to v . We define the neighborhood of v as N G ( v ) = ( S e ∈ E G ( v ) e ) \ { v } , the edge-degree of v as deg G ( v ) = | E G ( v ) | . We also define ∆( G ) = max { deg G ( v ) | v ∈ V ( G ) } . Given a F ⊆ E ( G ), wedefine G \ F = ( V ( G ) , E ( G ) \ F ).Given a tree T and two vertices a, b ∈ V ( T ) we define aT b as the path of T connecting a and b . Let G be a graph and let S , S ⊆ V ( G ) where S ∩ S = ∅ . We define E G ( S , S ) = { e ∈ E ( G ) | e ∩ V = ∅ and e ∩ V = ∅} . A cut of a graph G is any bipartition ( X, X ) of its vertices. The edges of a cut (
X, X ) is the set E ( X, X ) while the size of (
X, X ) is equal to | E ( X, X ) | . Given two distinct vertices x and y of G ,an ( x, y ) -cut of G is a cut ( X, X ) of G such that x ∈ X and y ∈ X .We define the function ρ : 2 V ( G ) → N such that ρ ( X ) = | E G ( X, V ( G ) \ X ) | . It is easy to seethat ρ is a submodular function, ie., ∀ X, Y ∈ V ( G ) ρ ( X ∩ Y ) + ρ ( X ∪ Y ) ≤ ρ ( X ) + ρ ( Y ) . (1)Given a graph G and two distinct x, y ∈ V ( G ), we call an ( x, y ) - s -path every s -path in G starting from x and finishing on y . We also use the term ( x, y ) -path as a shortcut for ( x, y )- ∞ -path.We define the function cut G,s : (cid:0) V ( G )2 (cid:1) → N ≥ so that cut G,s ( x, y ) is equal to the minimum sizeof a F ⊆ E ( G ) such that G \ F does not contain any ( x, y )- s -path. The complexity of computing cut G,s ( x, y ) is provided by the next proposition (see [3, 16, 19]). Proposition 2.1. If s ∈ { , , ∞} , then the problem that, given a graph G , a k ∈ N , and twodistinct vertices a and b of G , asks whether cut G,s ( a, b ) ≤ k is polynomially solvable, while it is NP -complete if s ∈ N ≥ . Graph searching and s -edge-degeneracy The study of graph searching parameters is an active field of graph theory. Several important graphparameters have their search-game analogues that provide useful insights on their combinatorialand algorithmic properties. (For related surveys, see [1, 2, 4, 11, 12].)We introduce a graph searching game, where the opponents are a group of cops and a robber.In this game, the cops are blocking edges of the graph, while the robber resides on the vertices .The first move of the game is done by the robber, who chooses a vertex to occupy. Then, the gameis played in rounds. In each round, first the cops block a set of edges and next the robber moves toanother vertex via a path consisting of at most s unblocked edges. The robber cannot stay put andhe/she is captured if, after the move of the cops, all the edges incident to his/her current locationare blocked. Both cops and robbers have full knowledge of their opponent’s current position andthey take it into consideration before they make their next move. We next give the formal definitionof the game.The game is parameterized by the speed s ∈ N + ≥ of the robber. A search strategy on G for thecops is a function f : V ( G ) → E ( G ) that, given the current position x ∈ V ( G ) of the robber in theend of a round, outputs the set f ( v ) of the edges that should be blocked in the beginning of thenext round. The cost of a cop strategy f is defined as cost ( f ) = max {| f ( v ) | | v ∈ V ( G ) } , i.e., themaximum number of edges that may be blocked by the robbers according to f .An escape strategy on G for the robber is a pair R = ( v start , g ) where v start is the vertex ofrobber’s first move and g : 2 E ( G ) × V ( G ) → V ( G ) is a function that, given the set F of blockededges in the beginning of a round and the current position x of the robber, outputs the vertex u = g ( F, v ) where the robber should move. Here the natural restriction for g is that there is an s -path from v to u in G \ F . Clearly, if F is the set of edges that are incident to v , then g ( F, v )should be equal to v and this expresses the situation where the robber is captured.Let f and R = ( v start , g ) be strategies for the cop and the robber respectively. The gamescenario generated by the pair ( f, R ) is the infinite sequence v , F , v , F , v , . . . , where v = v start and for every i ∈ N ≥ , F i = f ( v i − ) and v i = g ( F i , v i − ). If v i = v i − for some i ∈ N ≥ , then ( f, R )is a cop-winning pair, otherwise it is a robber-winning pair.The capture cost against a robber of speed s in a graph G , denoted by cc s ( G ) is the minimum k for which there is a cop strategy f , of cost at most k , such that for every robber strategy R , ( f, R )is a cop-winning pair. s -edge-degeneracy s -edge-degeneracy. Let G be a graph, x ∈ V ( G ), S ⊆ V ( G ) \ { x } , and s ∈ N + ≥ . We say that aset A ⊆ E ( G ) is an ( s, x, S ) -edge-separator if every s -path of G from x to some vertex in S , containssome edge from A . We define supp G,s ( x, S ) to be the minimum size of an ( s, x, S )-edge-separatorin G .Let G be a graph and let L = h v , . . . , v r i be a layout (i.e. linear ordering) of its vertices. Givenan i ∈ [ r ] , we denote L ≤ i = h v , . . . , v i i . Given an s ∈ N + ≥ , we define the s -edge-support of a vertex v i in L as supp G,s ( v i , L ≤ i − ). The s -edge-degeneracy of L , is the maximum s -edge-support of avertex in L . The s -edge-degeneracy of G , denoted by δ s e ( G ) is the minimum s -edge-degeneracy overall layouts of G . 4 k, s ) -edge-hide-outs. Let s ∈ N + ≥ and k ∈ N . A ( k, s ) -edge-hide-out in a graph G is any set R ⊆ V ( G ) such that, for every x ∈ R , supp G,s ( x, R \ { x } ) ≥ k . A ( k, s ) -edge-hide-out S is maximal there is no other ( k, s ) -edge-hide-out S with S (cid:40) S . It is easy to verify that every graph containsa unique maximal ( k, s )-edge-hide-out.( k, s )-edge-hide-outs can be seen as obstructions to small s -edge-degeneracy. In particular weprove the following min-max theorem, characterizing the search game that we defined in Subsec-tion 3.1. Theorem 3.1.
Let G be a graph and let s ∈ N + ≥ and k ∈ N . The following three statements areequivalent.(1) cc s ( G ) ≤ k , i.e., there is a cop strategy f on G of cost less than k , such that for every robberstrategy R on G , ( f, R ) is cop-winning.(2) G has no ( k + 1 , s ) -edge-hide-out.(3) δ s e ( G ) ≤ k .Proof. ( ⇒ (2) . We prove that the negation of (2) implies the negation of (1) . Suppose that S is a ( k + 1 , s )-edge-hide-out of G . We use S in order to build an escape strategy R = ( v start , g )on G as follows: Let v start be any vertex in S . Let now v ∈ S and F ∈ E ( G ) . If | F | > k , then g ( v, F ) = v . We next define g ( v, F ) for every F ∈ E ( G ) ≤ k . As S is a ( k + 1 , s )-edge-hide-out of G , we know that supp G,s ( v, S \ { v } ) ≥ k + 1, therefore there is an s -path from v to some vertex u ∈ S \ { v } that avoids all edges in F . We define g ( v, F ) = u . Notice now that if f is a cop strategyon G of cost at most k , and v , F , v , F , v , . . . , is the game scenario generated by the pair ( f, R ),then v i − = v i for every i ∈ N ≥ . This means that R is a robber-winning strategy against any copstrategy of cost at most k , therefore cc s ( G ) ≥ k + 1.( ⇒ (3) . Let n = | V ( G ) | . As G has no ( k +1 , s )-edge-hide-out, it follows that for every R ⊆ V ( G )there is a vertex v ∈ R , such that supp G,s ( v, R \{ v } ) ≤ k . We pick such a vertex for every R ⊆ V ( G )and we denote it by v ( R ). We now set V n = V ( G ), v n = v ( V n ), and for i ∈ h n − , . . . , i we set V i = V i +1 \ { v i +1 } , v i = v ( V i ). We now set L = h v , . . . , v n i and observe that for every i ∈ [ n ], supp G,s ( v i , L ≤ i − ) = supp G,s ( v i , V i − ) ≤ k . Therefore, the s -edge-degeneracy of L is at most k ,hence δ s e ( G ) ≤ k .( ⇒ (1) . Suppose now that L = h v , . . . , v n i is a layout of V ( G ) such that, for every i ∈ [ n ], supp G,s ( v i , L ≤ i − ) ≤ k . We use L to build a cop strategy f : V ( G ) → E ( G ) as follows. Let i ∈ [ n ]and let F i be an ( s, v i , L ≤ i − )-edge-separator of G . We define f by setting f ( v i ) = F i . This meansthat if at some point the robber occupies vertex v i , then there is no s -path in G \ F i from v i to L ≤ i − . As a consequence of this, no matter what the robber strategy R = ( v start , g ) is, it shouldhold that g ( v i , F i ) ∈ L ≥ i . Therefore if x , F , x , F , x , . . . , is the game scenario generated by thepair ( f, R ), then x i = x i − for some i < n . s -edge-degeneracy, for distinct values of s We now combine Proposition 2.1 with the min-max theorem of the previous subsection in orderto identify the computational complexity of δ s e for different values of s . Our main result is thefollowing. 5 heorem 3.2. If s ∈ { , , ∞} , then the problem that, given a graph G and a k ∈ N , asks whether δ s e ( G ) ≤ k , is polynomially solvable, while it is NP -complete if s ∈ N ≥ .Proof. Notice first that checking whether δ s e ( G ) ≤ k can be done by the algorithm check s -edgedegeneracy in Figure 1. Indeed, if the maximal ( k + 1 , s )-edge-hideout S is non-empty then theabove algorithm will report that δ s e ( G ) > k after visiting, in line 3, every vertex not in S , as, bythe maximality of S , for every S (cid:41) S there is a vertex x ∈ S \ S where supp G,s ( x, S \ { x } ) ≤ k .On the other hand, if S is empty, then the procedure will produce a layout L = h v , . . . , v n i with s -edge-degeneracy at most k . Algorithm check s -edge degeneracy Input: a graph G and an integer k ∈ N ≥ . Output: a report on whether δ s e ( G ) ≤ k .1. n ← | V ( G ) | , S ← V ( G ).2. for i = n, . . . , x ∈ S with supp G,s ( x, S \ { x } ) ≤ k then v i ← x ,else report that “ δ s e ( G ) > k ” and stop // S is the maximal ( k + 1 , s )-edge-hideout of G ,witnessing that δ s e ( G ) > k , because of Theorem 3.1.//4. S ← S − v i .5. Output “ δ s e ( G ) ≤ k , witnessed by layout L = h v , . . . , v n i . ” Figure 1: An algorithm checking whether δ s e ( G ) ≤ k .Clearly, check s -edge degeneracy runs in polynomial time if checking whether supp G,s ( x, S \{ x } ) ≤ k can be done in polynomial time, which is equivalent to checking whether cut G ,s ( x, x ) ≤ k where G is the graph obtained by G after we identify all vertices of S \ { x } to a single vertex x .As this is possible for s ∈ { , , ∞} , due to Proposition 2.1, the polynomial part of the theoremfollow.It now remains to prove that checking whether δ s e ( G ) ≤ k is an NP -hard problem when s ∈ N ≥ .For this we will reduce the problem of checking whether cut G,s ( a, b ) ≤ k to the problem of checkingwhether δ s e ( G ) ≤ k and the result will follow from the hardness part of Proposition 2.1.Let T s = ( G, a, b, k ) be a quadruple where G is a graph on n vertices, k ∈ N ≥ , and a , b twodistinct vertices of G . We construct the graph G T s as follows: Take k + n + 1 copies G , . . . , G k + n +1 of G and identify all a ’s of these copies to a single vertex that we call again a , while we set B := { b , . . . , b k + n +1 } where b i is the copy of b in G i . Next, we add n new vertices C = { c , . . . , c n } and, for every ( i, j ) ∈ [ n ] × [ k + n + 1], we add the edge e i,j = c i b j . The construction of G T s iscompleted by subdividing each edge e i,j s − i, j ) ∈ [ n ] × [ k + n + 1], we denote by P i,j the ( c i , b j )- s -path that replaces e i,j afterthis subdivision. Also we set P j = { P i,j | i ∈ [ n ] } , for j ∈ [ k + n + 1] , Q i = { P i,j | j ∈ [ k + n + 1] } , for i ∈ [ n ] , and P = S j ∈ [ k + n +1] P j .For the correctness of the reduction, it remains to prove the following. δ s e ( G T s ) ≤ k + n ⇐⇒ cut G,s ( a, b ) ≤ k (2)6e first claim that, for every j ∈ [ k + n + 1], cut G,s ( a, b ) = cut G T s ,s ( a, b j ) . (3)To see (3) observe that none of the ( b j , a )- s -paths of G T s contains any vertex outside G j , therefore cut G T s ,s ( a, b j ) = cut G j ,s ( a, b j ) = cut G,s ( a, b ).We first prove the ( ⇒ ) direction of the (2). For this we assume that cut G,s ( a, b ) ≥ k + 1and we show that G T s contains a ( k + n + 1 , s )-edge-hide-out, which, by Theorem 3.1, yields δ s e ( G T s ) ≥ k + n + 1. We claim that S := C ∪ B ∪ { a } is a ( k + n + 1 , s )-edge-hide-out of G T s .As cut G,s ( a, b ) ≥ k + 1 ≥
1, we know that for each j ∈ [ k + n + 1] there is a ( b j , a )- s -path, say R j , in G T s whose internal vertices are not vertices of any path in P . Moreover, every two paths in R := { R j | j ∈ [ k + n + 1] } have only one vertex, that is a in common. The fact that |R| = k + n + 1implies that cut G T s ,s ( a, B ) ≥ k + n + 1. Therefore, as cut G T s ,s ( a, S \ { a } ) ≥ cut G T s ,s ( a, B ) , wehave that cut G T s ,s ( a, S \ { a } ) ≥ k + n + 1 . (4)Consider now the vertex b j , for some j ∈ [ k + n + 1], and notice that that cut G T s ,s ( b j , W ∪{ a } ) ≥ cut G T s ,s ( a, b j ) + |P j | . Combining this with (3) and the fact that |P j | = n , we obtain that cut G T s ,s ( b j , W ∪ { a } ) ≥ cut G,s ( a, b ) + n ≥ k + n + 1. As cut G T s ,s ( b j , S \ { b j } ) ≥ cut G T s ,s ( b j , W ∪{ a } ) , we have that ∀ j ∈ [ k + n + 1] cut G T s ,s ( b j , S \ { b j } ) ≥ k + n + 1 . (5)Consider now the vertex c i , for some i ∈ [ n ]. Notice that cut G T s ,s ( c i , B ) ≥ |Q i | = k + n + 1.As cut G T s ,s ( c i , S \ { c i } ) ≥ cut G T s ,s ( c i , B ) we obtain that ∀ i ∈ [ n ] cut G T s ,s ( c i , S \ { c i } ) ≥ k + n + 1 . (6)It now follows from (4), (5), and (6), that S is an ( k + n + 1 , s )-edge-hide-out of G T s , as required.We now prove the ( ⇐ ) direction of (2). The assumption that cut G,s ( a, b ) ≤ k implies that cut G T s ,s ( a, b j ) ≤ k , because of (3). Therefore there is a set F j of edges in G i that blocks every( b j , a )- s -path of G T s .Let L = h v , . . . , v ‘ i be any layout of the vertices of G T s where L ≤ k +2 n +2 = h a, c , . . . , c n , b , . . . , b k + n +1 i (7)In order to prove that δ s e ( G T s ) ≤ k + n it suffices to show that, for each h ∈ [ ‘ ], supp G,s ( v h , L ≤ h − ) ≤ k + n. (8)Notice that the vertices of L ≤ k +2 n +2 are the vertices of S = W ∪ B ∪ { a } . As each other vertex v ∈ V ( G ) \ S , has degree at most n − G T s , we directly have that (8) holds when h ∈ [ k +2 n +3 , ‘ ].Let now v h = b j for some j ∈ [ k + n + 1]. Let F ∗ j be the edges incident to b j that are edges of thepaths in P j . Observe that F j ∪ F ∗ J blocks in G T s all the s -paths from L ≤ h − to b j . As all the edgesin F j ∪ F ∗ J have some endpoint in L ≥ h and | F j | + | F ∗ j | ≤ k + n , we conclude that (8) holds when h ∈ [ n + 2 , k + 2 n + 2]. Let now v h = c i , i ∈ [ n ]. Notice that the distance in G T s between c i andany vertex in { a } ∪ ( W \ { c i } ) is bigger than s , therefore supp G,s ( v h , L ≤ h − ) ≤ | F j | + | F ∗ j | ≤ k + n and (8) holds when h ∈ [2 , n + 1]. Finally (8) holds trivially when h = 1. This completes the proofof (2), and the theorem follows. 7 A structural theorem for edge-admisibility
This section is dedicated to the statement and proof of our structural characterization for δ ∞ e . Edge-admissibility
The ∞ -admissibility of a graph G is the minimum k for which there existsa layout L = h v , . . . , v n i of V ( G ) such that for every i ∈ [ n ] there are at most k vertex-disjoint,except for v i , paths from v i to L ≤ i − in G . If in this definition we replace “vertex-disjoint” by “edge-disjoint” (and we obviously drop the exception of v i ) we have an edge analogue of the admissibilityinvariant that, because of Menger’s theorem is the same invariant as δ ∞ e . This encourages us toalternatively refer to δ ∞ e ( G ) as the ∞ -edge-admissibility of the graph G .The purpose of this section is to give a structural characterization for graphs of bounded edge-admissibility. For this we need first a series of definitions. Immersions.
Given a graph G and two incident edges e and f of G (i.e., edges with a commonendpoint) the result of lifting e and f in G is the graph obtained from G after removing e and f and then adding the edge formed by the symmetric difference of e and f . We say that a graph H isan immersion of a graph G , denoted by H ≤ G , if a graph isomorphic to H can be obtained fromsome subgraph of G after a series of liftings of incident edges. Given a graph H , we define the classof H -immersion free graphs as the class of all graphs that do not contain H as an immersion. Edge sums.
Let G and G be graphs, let v , v be vertices of V ( G ) and V ( G ) respectivelysuch that k = deg G ( v ) = deg G ( v ), and consider a bijection σ : E G ( v ) → E G ( v ), where E G ( v ) = { e i | i ∈ [ k ] } . We define the k -edge sum of G and G on v and v , with respect to σ ,as the graph G obtained if we take the disjoint union of G and G , identify v with v , and then,for each i ∈ { , . . . , k } , lift e i and σ ( e i ) to a new edge e i and remove the vertex v . We say that G is a ( ≤ k ) -edge sum of G and G if either G is the disjoint union of G and G or there is some k ∈ [ k ], two vertices v and v , and a bijection σ as above such that G is the k -edge sum of G and G on v and v , with respect to σ . v v G G Figure 2: The graphs G and G and the graph created after the edge-sum of G and G .Let G be some graph class. We recursively define the ( ≤ k ) -sum closure of G , denoted by G ( ≤ k ) ,as the set of graphs containing every graph G ∈ G that is the ( ≤ k )-edge sum of two graphs G and G in G where | V ( G ) | , | V ( G ) | < | V ( G ) | .A graph G is almost k -bounded edge-degree if all its vertices, except possibly from one, haveedge-degree at most k . We denote this class of graphs by A k .The rest of this section is devoted to the proof of the the following result.8 heorem 4.1. For every graph G and k ∈ N ≥ , if G has edge-admissibility at most k , then G can be constructed by almost k -bounded edge-degree graphs after a series of ( ≤ k ) -edge sums, i.e., G ∈ A ( ≤ k ) k . Conversely, for every k ∈ N ≥ , every graph in A ( ≤ k ) k has edge-admissibility at most k − . θ k -immersion free graphs Recall that given a k ∈ N ≥ , θ k is the graph with two vertices and k parallel edges betweenthem. In this subsection we prove that θ k -immersion free graphs are exactly the graphs in A ( ≤ k ) k (Theorem 4.7).We need some more definitions in order to translate edge-sums to their decomposition equivalentthat will be more easy to handle. Tree-partitions. A tree-partition of a graph G is a pair D = ( T, B ) where T is a tree and B = { B t | t ∈ V ( T ) } is a near-partition of V ( G ). We refer to the sets in B as the bags of D . Givena tree-partition D = ( T, B ) of G and an edge e ∈ E ( T ), we define cross D ( e ) = E G ( V , V ), where V i = S t ∈ V ( T i ) B t , for i ∈ [2] and T and T are the two connected components of T \ e . tt t t T T T z
544 44 44 455 5 43 3 z z Z t Figure 3: A graph G , a tree-partition of G with adhesion 3, and the torso Z t of the vertex t .For each t ∈ V ( T ), we define the t -torso of D as follows: Let T , . . . , T q t be the connectedcomponents of T \ t and let t , . . . , t q t be the neighbors of t in T such that t i ∈ V ( T i ). We set¯ B i = S h ∈ V ( T i ) B h , for i ∈ [ q t ]. Ν ext, we define the graph Z i as the graph obtained from G if, forevery i ∈ [ q t ] , we identify all the vertices of ¯ B i to a single vertex z i (maintaining the multiple edgescreated after such an identification). We call Z t the t -torso of D or, simply a torso of D . We callthe new vertices z , . . . , z q t satellites of the torso Z t . For each i ∈ [ q t ], we say that z i represents thevertex t i in T and subsumes the connected component T i of T \ t . For an example of a tree-partition,see Figure 3. 9et D = ( B , T ) be a tree-partition of a graph G . The adhesion of D = ( T, B ) is max {| cross D ( e ) | | e ∈ E ( T ) } (the adhesion of the tree-partititon of Figure 3 is 3). The strength of D = ( T, B ) ismin { ∆( Z t ) | t ∈ V ( T ) } (in the tree-partititon of Figure 3 the red numbers are the values of ∆( Z t )for each node of the tree T ).Observe that if D has strength at least k + 1, then every torso of D contains a vertex of degreeat least k + 1.Notice that each graph G , where ∆( G ) ≤ k , has a tree-partition ( T, B ) where both adhesionand strength are at most k : let T be a star with center r and | V ( G ) | leaves ‘ , . . . , ‘ | V ( G ) | , considera numbering v , . . . , v | V ( G ) | of V ( G ), and then set B r = ∅ , while B ‘ i = { v i } , i ∈ [ | V ( G ) | ].The next observation follows directly from the definitions and provides a “translation” of edge-sums in terms of tree-partitions. Observation . Let G be a graph class and let k ∈ N . The class G ( ≤ k ) contains exactly the graphsthat have a tree-partition of adhesion at most k whose torsos are graphs in G . Lemma 4.3.
Let k ∈ N ≥ and let G be a graph and D = ( T, B ) be a tree-partition of G of adhesionat most k . If θ k +1 ≤ G , then there is a t ∈ V ( T ) such that θ k +1 ≤ Z t .Proof. Observe that if θ k +1 ≤ G ,then there are two vertices x and y in G that are connected by k + 1 pairwise edge-disjoint ( x, y )-paths, P , . . . , P k +1 in G . As D has adhesion at most k , there issome t ∈ V ( T ) such that x, y ∈ B t . Let T , . . . , T q t be the connected components of T \ t and let z , . . . , z q t be the satelites of the t -torso Z t of D . Let i ∈ [ k ] and notice that, among the edges ofthe ( x, y )-path P i , those missing from Z t are those that do not have endpoints in B t . Notice alsothat for every j ∈ [ q t ] the edges of P i with both endpoints in S t ∈ V ( T j ) B t appear as consecutiveedges in P i . We now contract each such set of edges to the vertex z j for each j ∈ [ q t ] and observethat the resulting path P i is a path of Z t . Observe that P , . . . , P k +1 are pairwise edge-disjoint( x, y )-paths of Z t and we conclude that θ k +1 ≤ Z t as required.Let D = ( T, B ) be a tree-partition of a graph G and k ∈ N ≥ . We say that a torso Z t of D is • k -splitable : if it contains a cut ( X, X ) of size smaller than or equal to k where both X and X contain some vertex of degree at least k + 1. • k -overloaded : if at least two of its vertices have degree at least k + 1.Given a tree-partition D = ( T, B ), we define w ( D ) = X t ∈ V ( T ) ( s D ( t ) − s D ( t ) is the number of vertices in B t that have degree at least k + 1. Observation . Let G be a graph, k ∈ N ≥ , and D be a tree-partition of G that has strength atleast k + 1. Then w ( D ) > k -overloaded.Given a k ∈ N ≥ , we say that a tree-partition D = ( B , T ) is k -tight if, its adhesion is at most k and its strength is at least k + 1. Lemma 4.5.
For every graph G and k ∈ N ≥ , if D is a k -tight tree-partition of G with a k -splittabletorso, then there is a k -tight tree-partition D of G where w ( D ) < w ( D ) . roof. Let Z t be a splittable torso of D and let L t = { z , . . . , z q t } be the satellite vertices of Z t . Wedenote by t , . . . , t q t be the vertices of T represented by z , . . . , z q t respectively. Also we denote by T , . . . , T q t the connected components of T \ t that are subsumed by z , . . . , z q t , respectively. Letalso Q t be the vertices of Z t that have degree at least k + 1. As the adhesion of D is at most k , itfollows that each vertex in L t has degree at most k . Therefore, Q t ⊆ B t .We now construct a tree-partition D of G . As Z t is k -splittable, there is a cut ( X, X ) of Z t , ofsize at most k and two vertices x, y where deg Z t ( x ) , deg Z t ( y ) ≥ k + 1, and x ∈ X and y ∈ X . Weset Q ( x ) t = Q t ∩ X and Q ( y ) t = Q t ∩ X and keep in mind that x ∈ Q ( x ) t and y ∈ Q ( y ) t . Note thatthere is a set I ⊆ [ q t ] such that X ∩ Z t = { z i | i ∈ I } and X ∩ Z t = { z i | i ∈ [ q t ] \ I } . We constructthe tree T as follows: we start from T \ t , then add two new adjacent vertices t x and t y , make t x adjacent with all vertices in { t i | i ∈ I } and make t y adjacent with all vertices in { t i | i ∈ [ q t ] \ I } .We also define B = { B h | h ∈ V ( T ) } such that if h ∈ V ( T ) \ { t } , then B h = B h . Finally, set B t x = B t ∩ X and B t y = B t ∩ X . Observe that • if e = t x t y , then | cross D ( e ) | = cut Z t ( X, X ) ≤ k , • if e = t y t i , i ∈ [ q t ] \ I , then | cross D ( e ) | = | cross D ( tt i ) | ≤ k , • if e = t x t i , i ∈ I , then | cross D ( e ) | = | cross D ( tt i ) | ≤ k , and • if e ∈ E ( T ) \ E ( T ) | , then | cross D ( e ) | = | cross D ( e ) | ≤ k .From the above, we deduce that the adhesion of D is at most k .Let now v ∈ V ( T ). As D has strength at least k + 1, then for each h ∈ V ( T ) \ { t } there is avertex in B h that has degree at least k + 1. This, together with the fact that x ∈ B t x and y ∈ B t y implies that D has strength at least k + 1. Therefore D is k -tight.We finally observe the following: • s D ( t x ) = | Q ( x ) t | , • s D ( t y ) = | Q ( y ) t | , and • if t ∈ V ( T ) \ { t x , t y } , then s D ( t ) = s D ( t )From the above, ( s D ( t x ) −
1) + ( s D ( t x ) −
1) = | Q t | − s D ( t ) − −
1, therefore w ( D ) < w ( D )as required.Given a tree T and two members a, a of E ( T ) ∪ V ( T ) we define aT a as the unique path in T starting from a and finishing on a . Also, given a vertex t ∈ V ( T ) we define its status of t as status ( T, t ) = X t ∈ V ( T ) | E ( tT t ) | , i.e., the sum of all the lengths of all the paths from t to the rest of the vertices of T .Let ( X, X ) and (
Y, Y ) be two cuts of a graph G . We say that the cuts ( X, X ) and (
Y, Y ) are parallel if X ⊆ Y , or X ⊆ Y , or X ⊆ Y , or Y ⊆ X . Lemma 4.6.
Let k ∈ N ≥ . If G is a θ k +1 -immersion free graph with at least one vertex of degreeat least k + 1 , Then G has a k -tight tree-partition where each torso has exactly one vertex of degreegreater than k . roof. Notice that G has at least one k -tight tree-partition that consists of a single bag containing allthe vertices of G . Among all k -tight tree-partitions of G , consider the set D containing every k -tighttree-partition of G , where w ( D ) takes the minimum possible value, say ‘ . From Observation 4.4 itis enough to prove that ‘ = 0, i.e., the tree-partitions in D contain no k -overloaded torsos. Assume,towards a contradiction, that ‘ >
0. Consider two vertices x and y , of G each of degree at least k + 1, that belong to the same bag of some tree-partition of D . Among all tree-partitions in D containing x, y in the same bag, say B t , we choose D = ( T, B ) to be one where status ( T, t ) isminimized.As θ k +1 G , the graph G contains some ( x, y )-cut ( X, X ) of size at most k . Let S x,y be theset of all such cuts.We say that an edge e ∈ E ( T ) is crossed by ( X, X ) if the cut of G corresponding to cross D ( e )and the cut ( X, X ) are not parallel. As both x and y have degree at least k + 1, there should betwo edges e x and e y in cross D ( e ) such that e x ⊆ X and e y ⊆ X .Let ( X, X ) ∈ S x,y . Let e = t t be an edge of E ( T ) that is crossed by ( X, X ). We make theconvention that, whenever we consider such an edge, we assume that | E ( tT t ) | < | E ( tT t ) | , i.e., t is closer to t than t , in T . We say that such an edge e is ( X, X ) -extremal for ( T, t ) if there isno other edge e = e of T that is crossed by ( X, X ) and such that e ∈ E ( e T t ). We denote by extr ( X, X ) the set of edges of T that are ( X, X )-extremal for (
T, t ). Notice that extr ( X, X ) shouldbe non-empty, as otherwise (
X, X ) should induce a cut of Z t , therefore Z t would be k -splittableand this, due to Lemma 4.5, would contradict the minimality of w ( D ). We next define the cost ofthe cut ( X, X ) as cost
T,t ( X, X ) = X t t ∈ extr ( X,X ) | E ( tT t ) | . We now pick the ( x, y )-cut (
X, X ) ∈ S x,y as one of minimum possible cost, in other words, cost ( X, X ) = min { cost ( X , X ) | ( X , X ) ∈ S x,y } .Let e = t t be an ( X, X )-extremal edge of T . Let ( A, A ) be the cut of G whose edges are cross D ( e ) and w.l.o.g., we assume that x, y ∈ A . Recall that ρ ( X ) = ρ ( X ) ≤ k and ρ ( A ) = ρ ( A ) ≤ k. (9)We next claim that ρ ( A ∩ X ) = | E ( A ∩ X, A ∪ X ) | > k. (10)To see (10), notice that if this is not the case, then ( A ∩ X, A ∪ X ) ∈ S x,y , because x ∈ A ∩ X and y ∈ A ∪ X . Notice that if t = t , then extr ( A ∩ X, A ∪ X ) = extr ( X, X ) \{ t t } while if t ∗ is the uniqueneighbor of t in the path joining t and t , then extr ( A ∩ X, A ∪ X ) = extr ( X, X ) \ { t t } ∪ { t t ∗ } .In both cases, cost ( A ∩ X, A ∪ X ) = cost ( X, X ) −
1, a contradiction to the minimality of the choiceof (
X, X ). Working symmetrically on A , instead of A , it follows that ρ ( A ∩ X ) = | E ( A ∩ X, A ∪ X ) | > k. (11)By the submodularity of ρ , we have that ρ ( A ∩ X ) + ρ ( A ∪ X ) ≤ ρ ( X ) + ρ ( A ) . (12) ρ ( A ∩ X ) + ρ ( A ∪ X ) ≤ ρ ( X ) + ρ ( A ) . (13)12 ∩ XA ∩ XA ∩ X A ∩ X XXt AA xyt (cid:48)(cid:48) t (cid:48) Figure 4: A visualization of the proof of Lemma 4.6.Combining now (9), (10), and (12) and (9), (11), and (13) we have that ρ ( A ∪ X ) ≤ k and ρ ( A ∪ X ) ≤ k which can be rewritten ρ ( A ∩ X ) ≤ k and ρ ( A ∩ X ) ≤ k. (14)Note that the vertices of B t that have degree at least k + 1 should all be in exactly one of A ∩ X and A ∩ X . Indeed, if this is not correct, then Z t should be k -splittable and this, due to Lemma 4.5,would contradict the minimality of w ( D ). W.l.o.g. we assume that Q = B t ∩ A ∩ X contains onlyvertices of degree at most k .Let z , . . . , z q t be the satellites of Z t and let t i be the vertex of T represented by z i , i ∈ [ q t ],assuming, w.l.o.g., that z represents t in T (that is t = t ). Let also T i be the connectedcomponent of T \ t subsumed by z i , for i ∈ [ q t ]. As t t ∈ extr ( X, X ), there is some non-empty I ⊆ [2 , q t ] such that [ i ∈ I [ s ∈ V ( T i ) B s = ( A ∩ X ) \ B t and [ i ∈ [2 ,q t ] \ I [ s ∈ V ( T i ) B s = ( A ∩ X ) \ B t . (15)We now add the set Q to B t and remove it from B t , and also remove from T all edges in { t i t | i ∈ I } and add the edges { t i t | i ∈ I } to get T (in Figure 4, the new edge is depicted bythe dashed edge). Observe that D = ( T , B ) is a tree-partition of G with adhesion at most k andwhere all its nodes contain some vertex of degree at least k + 1. Therefore D is k -tight. Noticethat, by the construction of T , status ( T , t ) < status ( T, t ) a contradiction to the minimality of status ( T, t ) in the choice of D = ( T, B ). Theorem 4.7.
For every graph G and k ∈ N , G is θ k +1 -immersion free if and only if G ∈ A ( ≤ k ) k .Proof. We prove first “only if” direction. If G has no vertices of degree at least k + 1, then G ∈ A k and the result follows trivially. If G has at least one vertex of degree at least k + 1, then, because13f Lemma 4.6, G has a k -tight tree-partition of adhesion at most k and whose torsos belong to A k .Then, from Observation 4.2, G ∈ A ( ≤ k ) k .We next prove the “if” direction. Suppose that G ∈ A ( ≤ k ) k , therefore, from Observation 4.2, G has a tree-partition D of adhesion at most k whose torsos are all in A k . As none of the torsos of D contains θ k +1 as an immersion, because of Lemma 4.3, the same holds for G and we are done.As mentioned by one of the reviewers, Theorem 4.7 can alternatively be proved by a suitableapplication of the theorem of Gomory and Hu [14] (see also [8] and [7]). In this subsection we prove that θ k +1 -immersion free graphs have edge-admissibility at most 2 k − Carving decompositions.
Given a tree T we denote by L ( T ) the set of all the vertices of T thathave degree at most 1 and we call them the leaves of T . A rooted tree is a pair T = ( T, r ) where T is a tree and r ∈ V ( T ). A binary rooted tree is a rooted tree T = ( T, r ) where all its non-leafvertices have exactly two children. If v ∈ V ( T ), we define descl T ( v ) as the set containing everyleave ‘ of T such that v ∈ V ( rT ‘ ).Let G be a graph and S ⊆ V ( G ). A rooted carving decomposition of G is a pair ( T , σ ) consistingof a rooted binary tree T = ( T, r ) and a function σ : V ( G ) → L ( T ). We stress that σ is not abijection, i.e., we permit many vertices of G to be mapped to the same leaf of T . The weight of avertex t in V ( T ) \ L ( T ) is defined as w ( t ) = | E G ( S , S ) | where S i = σ − ( descl T ( t i )) , i ∈ [2] and t , t are the children of t in T . For every edge e = tt of E ( T ), where t is a child of t , we define cut ( e ) as the set E G ( V , V ) where V = σ − ( descl T ( t ))and V = V ( G ) \ V . We also define the weight of e = tt as w ( e ) = | cut ( e ) | . Lemma 4.8.
Let G be a graph and k ∈ N ≥ . If θ k +1 (cid:2) G , then δ ∞ e ( G ) ≤ k − .Proof. We show that if G is θ k +1 -immersion free, then G cannot contain a (2 k, ∞ )-edge-hideoutand therefore, from Theorem 3.1, δ ∞ e ( G ) ≤ k −
1. Suppose to the contrary that S, | S | ≥
2, is a(2 k, ∞ )-edge-hideout of G . We build a rooted carving decomposition of G by applying the followingprecedure: Step 1 . Consider ( T , σ ) where T = ( T, v ), T consists of only one vertex, that is the root r , and σ ( v ) = r for all v ∈ V ( G ). Step 2 . Let ‘ be a vertex of T where | σ − ( ‘ ) ∩ S | ≥
2. If no such vertex exists, then stop . Step 3 . Pick, arbitrarily, two distinct vertices x and x in σ − ( ‘ ) ∩ S . Notice that G contains a( x , x )-cut ( X , X ) of at most k edges where x i ∈ X i , i ∈ [2], otherwise, from Menger’s theoremthere are k + 1 pairwise edge disjoint paths from x to x in G , which implies the existence of θ k +1 as an immersion in G , a contradiction. We now add in T two new vertices ‘ and ‘ make themthe children of ‘ and update σ so that the vertices in X i ∩ σ − ( ‘ ) are now mapped in ‘ i , i ∈ [2], i.e.we remove from σ ( t, σ − ( ‘ )) and we add ( t , X ∩ σ − ( ‘ )) and ( t , X ∩ σ − ( ‘ )). Step 4 . Go to
Step 2 . 14et ( T , σ ) be the rooted carving decomposition produced by the above procedure. By theconstruction of ( T , σ ), each vertex of T has weight at most k and for each leaf ‘ ∈ L ( G ), | σ − ( ‘ ) ∩ S | = 1. We construct a path P of T by applying the following procedure. Step 1 . Let P be the path of T consisting of r and one (arbitrarily chosen), say t , of the childrenof r (i.e., P is just an edge). Notice that w ( { r, t } ) = w ( r ) ≤ k ≤ k − k ≥ Step 2 . Let e be the the last edge of P (starting from r ) and let t be its endpoint that is also anendpoint of P (different than r ). If t is a leaf of T , then stop . Step 3 . Let t and t be the children of t and let e i = tt i , i ∈ [2]. We partition the edges of cut ( e )into two sets, namely F and F so that F i contains edges with an endpoint in descl T ( t i ) , i ∈ [2].Notice that cut ( e i ) = F i ∪ E G ( σ − ( descl T ( t )) , σ − ( descl T ( t ))) , therefore, for i ∈ [2], w ( e i ) = | cut ( e i ) | = | F i | + | w ( t ) | . (16)As w ( e ) ≤ k −
1, one, say F , of F , F should have at most k − i = 1, we obtain that | w ( e ) | ≤ k − w ( t ) ≤ k −
1. We now extend P by adding in it the vertex t and the edge e and we update e := e . Step 4 . Go to
Step 2 .We just constructed a path P in T between r and a leaf of ‘ of T such that for every edge e ∈ E ( P ), w ( e ) ≤ k −
1. Notice that σ − ( ‘ ) contains exactly one vertex, say x , of S . Moreover, if f is the edge of T that is incident to ‘ , then ρ ( σ − ( ‘ )) = w ( f ) ≤ k −
1, as f is an edge of P (thelast one). This implies that there is a set of 2 k − x to S \ { x } .Therefore, supp G ( ∞ , x, S \{ x } ) ≥ k −
1, contradicting to the fact that S is a (2 k, ∞ )-edge-hideoutof G . Observation . If H and G are graphs then H ≤ G ⇒ δ ∞ e ( H ) ≤ δ ∞ e ( G ). Proof.
Suppose that H ≤ G and that k ≤ δ ∞ e ( H ). From Theorem 3.1 H contains a ( k, ∞ )-edge-hide-out S ⊆ V ( H ). Because of Menger’s theorem, for every vertex v ∈ S there are at least k + 1pairwise edge-disjoint paths from v to vertices of S \ { v } . Notice that these paths also exist in G as the “inverse” of the lift operation does not alter the paths from a vertex of S to the rest of thevertices of S . These paths, again using Menger’s theorem, imply that S is also a ( k + 1 , ∞ )-edge-hide-out of G , therefore, again from Theorem 3.1, k ≤ δ ∞ e ( G ).We are now ready to give the proof of Theorem 4.1. Proof of Theorem 4.1.
For the first part of the theorem, observe that δ ∞ e ( θ k +1 ) = k + 1, therefore,from Observation 4.9, θ k +1 (cid:2) G . Using now the “only if” direction of Theorem 4.7 we obtain that G ∈ A ( ≤ k ) k , as required.For the second part of the theorem, let G ∈ A ( ≤ k ) k , which by the “if” direction of Theorem 4.7implies that θ k +1 (cid:2) G . Using now Lemma 4.8, we conclude that δ ∞ e ( G ) ≤ k − Acknowledgements
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