A Linear Time Algorithm for Computing the Eternal Vertex Cover Number of Cactus Graphs
AA Linear Time Algorithm for Computing the Eternal Vertex Cover Numberof Cactus Graphs
Jasine Babu, Veena Prabhakaran, Arko Sharma
Indian Institute of Technology Palakkad, Kerala, India 678557
Abstract
The eternal vertex cover problem is a dynamic variant of the classical vertex cover problem. It is NP-hard tocompute the eternal vertex cover number of graphs and known algorithmic results for the problem are veryfew. This paper presents a linear time recursive algorithm for computing the eternal vertex cover numberof cactus graphs. Unlike other graph classes for which polynomial time algorithms for eternal vertex covernumber are based on efficient computability of a known lower bound directly derived from minimum vertexcover, we show that it is a certain substructure property that helps the efficient computation of eternalvertex cover number of cactus graphs. An extension of the result to graphs in which each block is an edge,a cycle or a biconnected chordal graph is also presented.
Keywords:
Eternal vertex cover, Cactus graphs, Linear time algorithm, Chordal graphs.
1. Introduction
Eternal vertex cover problem can be described in terms of a two player game played on any given graph G between an attacker who attacks the edges of G and a defender who controls the placement of a fixednumber of guards (say k ) on a subset of vertices of G to defend the attack [1]. In the beginning of the game,the defender chooses a placement of guards on a subset of vertices of G , defining an initial configuration .Subsequently, in each round of the game, the attacker attacks an edge of her choice. In response, the defenderhas to reconfigure the guards. The guards move in parallel. Each guard may be retained in its position ormoved to a neighboring vertex, ensuring that at least one guard from an endpoint of the attacked edge movesto its other endpoint. If the defender is able to do this, then the attack is said to be successfully defendedand the game proceeds to the next round with the attacker choosing the next edge to attack . Otherwise,the attacker wins. If the defender is able to keep defending against any infinite sequence of attacks on G with k guards, then we say that there is a defense strategy on G with k guards. In this case, the positionsof guards in any round of the game defines a configuration , referred to as an eternal vertex cover of G ofsize k . The set of all configurations encountered in the game defines an eternal vertex cover class of G . The eternal vertex cover number of a graph G , denoted by evc( G ) is the minimum integer k such that there is adefense strategy on G using k guards. It is clear from the description of the game that in any configuration,at least one of the endpoints of each edge must have a guard for defending attacks successfully. Hence, itis clear that each eternal vertex cover of G is a vertex cover of G and evc( G ) ≥ mvc( G ), the vertex covernumber of G .The decision version of the eternal vertex cover problem takes a graph G and an integer k as inputs andasks whether G has an eternal vertex cover of size at most k . Fomin et al. [2] showed that this problem is Email addresses: [email protected] (Jasine Babu), [email protected] (Veena Prabhakaran), [email protected] (Arko Sharma) There are two popular versions of the problem: one as described here and the second with an additional constraint thatnot more than one guard can be on any vertex in any configuration. The number of guards required to protect the graph maydiffer between these two versions. However, the results derived in this paper and those cited here hold in both the versions.
Preprint submitted to arXiv May 19, 2020 a r X i v : . [ c s . D M ] M a y P-hard and is in PSPACE. The problem was recently shown to be NP-Complete for internally triangulatedplanar graphs [3]. It is not yet known whether the problem is in NP for bipartite graphs [2]. Similarly,it is an open question whether computing the eternal vertex cover number of bounded treewidth graphs isNP-Hard or not [2]. The problem remains open even for graphs of treewidth two.Dynamic variants of other classical graph parameters like dominating set [4, 5, 6, 7] and independentset [8, 9] are also well studied in literature. Rinemberg et al. [6] recently showed that eternal dominationnumber of interval graphs can be computed in polynomial time. The relationship between eternal vertexcover number and other dynamic graph parameters was explored by Klostermeyer and Mynhardt [10] andAnderson et al. [11]. It is known that evc( G ) ≤ cvc( G ) + 1, where cvc( G ) is the minimum cardinality ofa connected vertex cover of G [1, 2]. Fomin et al. [2] showed that evc( G ) is at most twice the size of amaximum matching in G . It is also known that for any connected graph G , evc( G ) ≥ mvc X ( G ), wheremvc X ( G ) is the minimum cardinality of a vertex cover of G that contains all cut vertices of G [3].Precise formulae or algorithms for computing eternal vertex cover number are known for only a fewclasses of graphs. The initial work by Klostermeyer and Mynhardt gave explicit formulae for the eternalvertex cover number of trees, cycles and grids [1]. The eternal vertex cover number of a tree T is known tobe | V ( T ) | − | L ( T ) | + 1, where L ( T ) is the number of leaves of T . The eternal vertex cover number of a cycleon n vertices is (cid:100) n (cid:101) , which is equal to its vertex cover number. A polynomial time algorithm for computingeternal vertex cover number of chordal graphs has been obtained recently [3]. Parameterized algorithms forthe problem are discussed by Fomin et al. [2]. Araki et al. [12] discusses a polynomial time algorithm tocompute eternal vertex cover number of a simple generalization of trees.All the graph classes for which polynomial time algorithms for eternal vertex cover are known so farexhibit a common property. Recall that mvc X ( G ), which is the minimum cardinality of a vertex coverof G that contains all cut vertices of G is a lower bound to evc( G ). For any graph G that belong to theclasses given above, this lower bound is polynomial time computable and evc( G ) ∈ { mvc X ( G ) , mvc X ( G )+1 } .However, this is not true for cactus graphs, as shown in Section 6. Hence, we formulate a new lower boundingtechnique based on the substructure property (Definition 4) to handle cactus graphs.A cactus is a connected graph in which any two simple cycles have at most one vertex in common. Acactus is a generalization of cycles and trees in the following sense: if we contract every cycle in a cactus toa single vertex, the resultant graph is a tree. Cactus graphs have treewidth at most two. Many importantNP-hard combinatorial problems are polynomial time solvable on cactus graphs [13]. Since trees and cyclesare examples of cactus graphs, we know that there exist cactus graphs G for which mvc X ( G ) lower bound isclose to evc( G ). However, there are cactus graphs G for which the eternal vertex cover number is more than1 . X ( G ) (Section 6). Thus, the reason for efficient computability of eternal vertex cover numberfor this graph class is not just related to the lower bound mvc X ( G ).The lower bound argument for eternal vertex cover number of trees was obtained using a recursiveprocedure [1]. We show that by using a more sophisticated recursive method based on the substructureproperty of cactus graphs, we can device a new lower bounding strategy applicable for cactus graphs. Fromthis, we derive a formula for computing the eternal vertex cover number of a cactus G , in terms of eternalvertex cover number and some related parameters of certain edge disjoint subgraphs of G . The algorithmfor computing eternal vertex cover number and the other required parameters is a recursive procedure basedon this formula and it runs in time linear in the size of the input graph. Finally, we present an extension ofthe result to graphs in which each block is an edge, a cycle or a biconnected chordal graph. It is shown thatfor graphs of this class, eternal vertex cover number can be computed in quadratic time. In particular, thismethod implies a quadratic time algorithm for the computation of eternal vertex cover number of chordalgraphs, which is different from the algorithm given in [3].
2. Some Basic ObservationsDefinition 1.
Let G be a graph and S ⊆ V ( G ) . The minimum cardinality of a vertex cover of G that containsall vertices of S is denoted by mvc S ( G ) . The minimum integer k such that there is a defense strategy on G using k guards with all vertices of S being occupied in each configuration is denoted by evc S ( G ) . S = { v } , we use mvc v ( G ) and evc v ( G ) respectively instead of mvc S ( G ) and evc S ( G ). From theabove definition, it is clear that evc( G ) ≤ evc S ( G ). Definition 2. If G is a graph and x ∈ V ( G ) , we use G + x to denote the graph obtained by adding an additionalvertex which is made adjacent only to x . Observation 1. evc( G + x ) ≤ evc( G ) + 1 .Proof. Suppose evc( G ) = k . Let v be the vertex in V ( G + x ) \ V ( G ). There is a defense strategy on G + x using k + 1 guards, by using k guards to move exactly the same way as they would have been moved for defendingattacks on G and an additional k + 1 th guard to protect against attacks on the edge xv . If the edge xv isattacked when x is not occupied, then the additional guard will move from v to x . In the very next round,this guard moves back to v , irrespective of the edge attacked. It is easy to check that using this strategy,any attack on G + x can be defended. Definition 3 (x-components and x-extensions) . Let x be a cut vertex in a connected graph G and H be acomponent of G \ x . Let G be the induced subgraph of G on the vertex set V ( H ) ∪ { x } . Then, G is calledan x -component of G and G is called an x -extension of G . Definition 4 (Substructure property) . Let x be an arbitrary non-cut vertex of a graph G . If the followingis true for any arbitrary x -extension G (cid:48) of G , then G satisfies substructure property: • if evc( G + x ) ≤ evc x ( G ) , then in every eternal vertex cover C (cid:48) of G (cid:48) , the number of guards on V ( G ) isat least evc x ( G ) − and • if evc( G + x ) > evc x ( G ) , then in every eternal vertex cover C (cid:48) of G (cid:48) , the number of guards on V ( G ) isat least evc x ( G ) . Observation 2.
If a graph G satisfies substructure property and x is a non-cut vertex of G , then evc x ( G ) ≤ evc( G + x ) ≤ evc x ( G ) + 1 .Proof. Suppose G satisfies substructure property and x is a non-cut vertex of G . Let v be the vertex in V ( G + x ) \ V ( G ). In any eternal vertex cover class of G + x , there must be a configuration in which there isa guard on v . By substructure property, in this configuration, the number of guards in V ( G ) must be atleast evc x ( G ) −
1. Hence, the number of guards required on V ( G + x ) is at least evc x ( G ). Consequently,evc( G + x ) ≥ evc x ( G ). Further, by Observation 1, it follows that evc( G + x ) ≤ evc x ( G ) + 1. Definition 5.
Let G be a graph and x be a vertex of G . G is Type 1 with respect to x if evc( G + x ) = evc x ( G ) and G is Type 2 with respect to x if evc( G + x ) > evc x ( G ) . Remark 1.
From the above definition, a tree is Type 1 with respect to any of its vertices. An even cycleis Type 1 with respect to any of its vertices. An odd cycle is Type 2 with respect to any of its vertices. Acomplete graph on or more vertices is Type 2 with respect to any of its vertices. Remark 2.
By Observation 2, if G satisfies substructure property and x is a non-cut vertex of G , then G is either Type 1 or Type 2 with respect to x . Definition 6 ([3]) . Let x be a cut vertex of a connected graph G . The set of x -components of G will bedenoted as C x ( G ) . For i ∈ { , } , we define T i ( G, x ) to be the set of all x -components of G that are Type i with respect to x . Lemma 1.
Let G be a connected graph and x be a cut-vertex of G such that each x -component of G satisfiesthe substructure property. If all x -components of G are Type 2, then evc( G ) = evc x ( G ) = 1 + (cid:80) G i ∈C x ( G ) (evc x ( G i ) − . Otherwise, evc( G ) = evc x ( G ) = 2 + (cid:80) G i ∈ T ( G,x ) (evc x ( G i ) −
2) + (cid:80) G i ∈ T ( G,x ) (evc x ( G i ) − . roof. The first part is easy to see from the definition of the substructure property.To prove the second part, suppose H is an x -component of G which is Type 1 with respect to x . Let k = 1+ (cid:80) G i ∈ T ( G,x ) (evc x ( G i ) − (cid:80) G i ∈ T ( G,x ) (evc x ( G i ) − G ) cannot be less than k . Now, for contradiction, suppose evc( G ) = k . Let C be a minimum eternal vertex cover class G . Since there are only k guards on V ( G ) and each x -componentof G satisfies the substructure property, it can only be the case that in every configuration of C the numberof guards on an x -component G i is exactly evc x ( G i ) − G i is Type 1 with respect to x and it is exactlyevc x ( G i ) if G i is Type 2 with respect to x . Further, x must also be occupied in every configuration. However,to defend repeated attacks on edges of H which is a Type 1 x -component with respect to x , evc x ( H ) guardshave to be on V ( H ) in some configuration, if x is to be always occupied. Then, the total number of guardson V ( G ) must be more than k , a contradiction. Hence, evc( G ) ≥ k + 1.It is not difficult to show that evc x ( G ) ≤ k + 1. In every configuration, we will maintain the followinginvariants. • A guard will be kept on x . • There will be exactly evc x ( G i ) guards on each x -component of G that is Type 2 with respect to x suchthat the guards on V ( G i ) form an eternal vertex cover of G i . • In one of the Type 1 x -components G i of G (if exists), exactly evc x ( G i ) guards will be kept and inevery other Type 1 component G j of G , evc x ( G j ) − V ( G i ), for each x -component G i forms an induced configurationof an eternal vertex cover of ( G i ) + x .To defend an attack on an edge in a Type 2 component G i maintaining the invariants, it will be enoughto move guards only on V ( G i ). To defend an attack on an edge in a Type 1 component, it will be enoughto move guards in at most two x -components. If the edge attacked is in a Type 1 x -component G i withevc x ( G i ) − V ( G i ), then there is another Type 1 x -component G j with evc x ( G j ) guards on G j .Since G i and G j satisfy substructure property, it is not difficult to rearrange guards in V ( G i ) and V ( G j )such that the number of guards on V ( G i ) becomes evc x ( G i ) and that on V ( G j ) becomes evc x ( G j ) −
3. Computation of the Type of vertices
Lemma 1 indicates the possibility of a recursive method to compute eternal vertex cover number ofgraphs whose x -components satisfy the substructure property. However, this makes it necessary to alsocompute evc x ( G ) and the type of each x -component of G , with respect to x . Since a cut vertex of G is nota cut-vertex in its x -components, a general method to find the type of a graph with respect to any arbitraryvertex of the graph (including non-cut vertices) is necessary. This section addresses this issue systematically. Observation 3 (Type with respect to a cut vertex) . Let G be a connected graph and x be a cut-vertex of G such that each x -component of G satisfies the substructure property. Then, G is Type 1 with respect to x ifat least one of the x -components of G is Type 1 with respect to x . Otherwise, G is Type 2 with respect to x .Proof. Note that when a pendent edge xv is added to x , that edge is a Type 1 component with respect to x .Further, evc x of this x -component is 2. Using these facts in the expressions given in Lemma 1 immediatelyyields the observation. Observation 4.
Let G be a graph that satisfies substructure property and x be any vertex of G . If evc( G ) < evc x ( G ) , then G is Type 1 with respect to x .Proof. If evc( G ) < evc x ( G ), then by Observation 1, we have evc( G + x ) ≤ evc( G ) + 1 ≤ evc x ( G ). By Remark 2and Observation 3, evc( G + x ) ≥ evc x ( G ). Therefore, evc x ( G ) = evc( G + x ). From this, the observation follows.4he following observation is useful for deciding the type of a graph with respect to a pendent vertex. Lemma 2 (Type with respect to a pendent vertex) . Let G be a graph and x ∈ V ( G ) . Let H = G + x with v being the vertex in V ( H ) \ V ( G ) . Suppose each x -component of H satisfies substructure property. Then, H is Type 1 with respect to v if and only if G is Type 1 with respect to x .Proof. In any eternal vertex cover class of H with evc v ( H ) guards in which v is occupied in every config-uration , x must also be occupied in every configuration (otherwise, an attack on the edge vx cannot bedefended maintaining a guard on v ). Hence, the induced configurations on G define an eternal vertex coverclass of G in which x is occupied in every configuration. It follows that evc v ( H ) ≥ evc x ( G ) + 1. Moreover,from an eternal vertex cover class of G with x always occupied, we can get an eternal vertex cover classof H with v always occupied by placing an additional guard at v . Hence, evc v ( H ) ≤ evc x ( G ) + 1. Thus,evc v ( H ) = evc x ( G ) + 1. Note that, by Observation 1, evc( H + v ) ≤ evc( H ) + 1.First, suppose G is Type 1 with respect to x . Then, evc( H ) = evc x ( G ) and we get evc v ( H ) = evc x ( G ) +1 = evc( H ) + 1 = evc( H + v ), which means that H is Type 1 with respect to v .Now, suppose G is Type 2 with respect to x . Then, evc( H ) = evc x ( G ) + 1. Further, by Observation 3, if x is a cut vertex in G , then all the x -components of G are Type 2 with respect to x . Irrespective of whether x is a cut-vertex of G or not, by Lemma 1, evc( H + v ) = 2 + evc x ( G ). Hence, evc( H + v ) = evc v ( H ) + 1. Thus,if G is Type 2 with respect to x , then H is Type 2 with respect to v .Now we give some observations that are useful for deciding the type of a graph with respect to a degree-2vertex. Lemma 3.
Let G be any graph and suppose v is a degree- vertex in G such that its neighbors v , v arenot adjacent. Let G (cid:48) be the graph obtained by deleting v and adding an edge between v and v . Then, evc v ( G ) = evc( G (cid:48) ) + 1 .Proof. We first give a proof for the version of the problem where at most one guard is allowed on a vertex.Consider a minimum eternal vertex cover class C (cid:48) of G (cid:48) . Note that, at least one among v and v isoccupied in each configuration of C (cid:48) . Let C = { S (cid:48) ∪ { v } : S (cid:48) ∈ C (cid:48) } . It is easy to see that each S ∈ C is avalid vertex cover of G . If S (cid:48) , S (cid:48) are configurations in C (cid:48) obtainable from each other by a single step of validmovement of guards, it is also easy to verify that the corresponding configurations S (cid:48) ∪ { v } and S (cid:48) ∪ { v } of G are obtainable from each other by a single step of valid movement of guards. From this, it follows easilythat C is an eternal vertex cover class of G with v always occupied. Hence, evc v ( G ) ≤ evc( G (cid:48) ) + 1.The correspondence between configurations of G (cid:48) and G works in the reverse direction as well. Let C bean eternal vertex cover class of G in which v is permanently occupied. Let C (cid:48) = { S \ { v } : S ∈ C} . Anyconfiguration S in C must contain at least one of v and v . Therefore, the corresponding configuration S (cid:48) = S \ { v } is a vertex cover of G (cid:48) . Moreover, whenever the guards in G move from a configuration S to S in C via a reconfiguration of guards which involves moving the guard from v to v and v to v , we can simulatethe behavior in G (cid:48) by assuming that the guard on v is moving to v (similarly while moving from v to v and v to v ). This corresponds to a valid movement of guards in G (cid:48) from configuration S \ { v } to S \ { v } .From this, it follows easily that C (cid:48) is an eternal vertex cover class of G (cid:48) . Thus, evc( G (cid:48) ) ≤ evc v ( G ) − v ( G ) ≤ evc( G (cid:48) ) + 1. In the second part of theproof, if S is a configuration in C such that for each x ∈ V ( G ) there are exactly t x guards on x , then definethe corresponding configuration S (cid:48) of C (cid:48) to have t v + t v − v and for each x ∈ V ( G ) \ { v, v } , t x guards on x . This modification is sufficient to prove evc( G (cid:48) ) ≤ evc v ( G ) − Definition 7.
Let X be the set of cut vertices of a graph G . If B is a block of G , the set of B -componentsof G is defined as C B ( G ) = { G i : G i ∈ C x ( G ) for some x ∈ X ∩ V ( B ) and G i edge disjoint with B } . If P is In the version of the problem where more than one guard is allowed on a vertex, we still can assume without loss ofgenerality that v has only one guard in any configuration. This is because instead of placing more than one guard on v , it ispossible to place all but one of those guards on x . path in G , then the set of P -components of G is defined as C P ( G ) = { G i : G i ∈ C x ( G ) for some x ∈ X ∩ V ( P ) and G i edge disjoint with P } . Definition 8.
For a block B (respectively, a path P ) of connected graph G , the type of a B -component(respectively, P -component) is its type with respect to the common vertex it has with B (respectively, P ).For i ∈ { , } , we define T i ( G, B ) (respectively, T i ( G, P ) ) to be the set of all B -components (respectively, P -components) of G that are Type i . If B is a block (respectively, P is a path) of a connected graph G , such that all B -components (respectively, P -components) of G satisfy substructure property, then we can easily obtain a lower bound on the totalnumber of guards on (cid:83) G i ∈C B ( G ) V ( G i ) (respectively, on (cid:83) G i ∈C P ( G ) V ( G i )) in any eternal vertex cover of G or its extensions. The notation introduced below is to abstract this lower bound. Definition 9.
For a block B of connected graph G , we define χ ( G, B ) = | V ( B ) ∩ X | + (cid:88) G i ∈ T ( G,B ) x i ∈ V ( G i ) ∩ V ( B ) (evc x i ( G i ) −
2) + (cid:88) G i ∈ T ( G,B ) x i ∈ V ( G i ) ∩ V ( B ) (evc x i ( G i ) − Similarly, for a path P of connected graph G , we define χ ( G, P ) = | V ( P ) ∩ X | + (cid:88) G i ∈ T ( G,P ) x i ∈ V ( G i ) ∩ V ( B ) (evc x i ( G i ) −
2) + (cid:88) G i ∈ T ( G,P ) x i ∈ V ( G i ) ∩ V ( B ) (evc x i ( G i ) − . Remark 3. If B is a block (respectively, P is a path) of a connected graph G , such that all B -components(respectively, P -components) of G satisfy substructure property, then the total number of guards on (cid:83) G i ∈C B ( G ) V ( G i ) (respectively, on (cid:83) G i ∈C P ( G ) V ( G i ) ) in any eternal vertex cover of G or its extensions isat least χ ( G, B ) (respectively, χ ( G, P ) ). Definition 10 (Vertex bunch of a path) . Let P be a path in a connected graph G . The vertex set V ( P ) ∪ (cid:83) G i ∈C P ( G ) V ( G i ) is the vertex bunch of P in G , denoted by VB G ( P ) . Definition 11 (Eventful path) . Let G be a connected graph and X be the set of cut vertices of G . A path P in a graph G is an eventful path if • P is either an induced path in G or a path obtained by removing an edge from an induced cycle in G . • the endpoints of P are in X and • any subpath P (cid:48) of P with both endpoints in X has | V ( P (cid:48) ) \ X | even. Lemma 4.
Let G be a connected graph and let P be an eventful path in G . Let X be the set of cut verticesof G . If each P -component in C P ( G ) satisfies the substructure property, then in any eternal vertex coverconfiguration of G , the total number of guards on VB G ( P ) is at least | V ( P ) \ X | + χ ( G, P ) . Moreover, if VB G ( P ) (cid:54) = V ( G ) and the number of guards on VB G ( P ) is exactly equal to the above expression, then atleast one of the neighbors of the endpoints of P outside VB G ( P ) has a guard on it.Proof. Consider any subpath P (cid:48) of P such that both endpoints of P (cid:48) are in X and none of its intermediatevertices are from X . Let C be an eternal vertex cover configuration of G . Since P is eventful, | V ( P (cid:48) ) \ X | is even and in any vertex cover of G , at least | V ( P (cid:48) ) \ X | internal vertices of P (cid:48) must be present. Using thisalong with the substructure property of P -components proves the first part of the lemma.Now, suppose VB G ( P ) (cid:54) = V ( G ) and the number of guards in the configuration C on VB G ( P ) is exactlyequal to the expression given in the lemma. Now, for contradiction, let us assume that none of the neighborsof the endpoints of P outside VB G ( P ) has a guard in C . Consider an attack on an edge xv , where x is an6ndpoint of P and v is a neighbor of x outside VB G ( P ). To defend this attack, a guard must move from x to v . Note that, no guards can move to VB G ( P ) from outside VB G ( P ). Hence, while defending the attack,the number of guards on VB G ( P ) decreases at least by one. But, then the new configuration will violatethe first part of the lemma. Hence, it must be the case that at least one of the neighbors of the endpointsof P outside VB G ( P ) has a guard in C . Definition 12 (Maximal uneventful path) . Let G be a connected graph and let X be the set of cut verticesof G . A path P in G is a maximal uneventful path in G if • V ( P ) ∩ X = ∅• | V ( P ) | is odd and • P is a maximal induced path in G satisfying the above two conditions. The next lemma is applicable to any connected graph G that contains a block B which is a cycle suchthat all B -components satisfy the substructure property. Since each block of a cactus is either a cycle or anedge, this lemma will be useful for computing the eternal vertex cover number of cactus graphs. The proofof the lemma makes use of the fact that B can be partitioned into a collection of edge disjoint paths whichare either eventful paths or maximal uneventful paths. Lemma 5.
Let B be a cycle forming a block of a connected graph G and let X be the set of cut verticesof G . Suppose each B -component G i of G that belongs to C B ( G ) satisfies the substructure property. If T ( G, B ) = ∅ , then evc( G ) = (cid:108) | V ( B ) \ X | (cid:109) + χ ( G, B ) . Otherwise, evc( G ) = (cid:108) | V ( B ) \ X | +12 (cid:109) + χ ( G, B ) .Proof. Let | V ( B ) | = n and | X ∩ V ( B ) | = k . For each B -component G i of G that belongs to C B ( G ), let x i be the vertex that G i has in common with B . Let C be a minimum eternal vertex cover of G . Note that thecondition stated in Lemma 4 has to simultaneously hold for all subpaths of the cycle that are eventful in G .Let l ≥ B and let P , P , . . . , P l be these listed in thecyclic order along B . To protect the edges within each P i , V ( P i ) should contain at least mvc( P i ) = (cid:98) | V ( P i ) | (cid:99) guards. If there are exactly (cid:98) | V ( P i ) | (cid:99) guards on V ( P i ), the end vertices of P i are not occupied and alternatevertices in P i are occupied by guards. • Case 1 : T ( G, B ) = ∅ a. when n − k is odd.In this case, from Definition 11, it follows that l is odd. – Suppose l = 1. Let P be the subpath obtained from B by deleting the edges of P . It iseasy to see that P is an eventful path. If V ( P ) contains only (cid:98) | V ( P ) | (cid:99) guards, then the endvertices of P are not occupied by guards and the condition stated in Lemma 4 cannot holdfor P . Therefore, the lemma holds when l = 1. – Suppose l >
1. Then, if V ( P i ) and V ( P i +1 ) (+ is mod l ) respectively contains only (cid:98) | V ( P i ) | (cid:99) and (cid:98) | V ( P i +1 ) | (cid:99) guards and the vertex bunch of the path P between the last vertex of P i andthe first vertex of P i +1 contains exactly as many guards as mentioned in the first part ofLemma 4, the condition stated in the second part of Lemma 4 cannot hold for P . Since thisis true for all 1 ≤ i ≤ l , and the condition stated in Lemma 4 has to simultaneously holdfor all subpaths of the cycle that are eventful in G , a simple counting argument shows thatnumber of guards in C should be at least (cid:6) n − k (cid:7) + χ ( G, B ).Thus, we know that evc( G ) is at least the expression given above. If T ( G, B ) = ∅ , it iseasy to show that these many guards are also sufficient. The guards on V ( B ) can defendany attack on edges of B while keeping X ∩ V ( B ) always occupied. Attacks on edges of B -components can be handled, maintaining x i always occupied and having exactly evc x i ( G i )guards on each B -component G i . 7. When n − k is evenIn this case, l is also even. A similar counting argument will show that evc( G ) is given by theexpression in the statement of the lemma. • Case 2 : T ( G, B ) (cid:54) = ∅ a. when n − k is even.Suppose there are exactly n − k + χ ( G, B ) guards on G . Then, it is not difficult to see that in orderto satisfy the condition stated in Lemma 4 for all eventful subpaths of the cycle, the configurationof guards should be such that – All cut vertices have guards. – while going around the cycle B (discarding the cut vertices), non-cut vertices are alternatelyguarded and unguarded. – each G i ∈ T ( G, B ) contains exactly evc x i ( G i ) − – each G i ∈ T ( G, B ) contains exactly evc x i ( G i ) guards.Now, consider a Type 1 B -component G i . By the conditions listed above, x i must be occupiedin every configuration. Hence, there is a sequence of attacks on G i that would eventually lead toa configuration with at least evc x i ( G i ) guards on V ( G i ) to defend the attack. However, in thatconfiguration, the conditions listed above will not be satisfied. Hence, we need at least one moreguard. We argue below that with one more guard, we can maintain the following invariants inevery configuration: – All cut vertices have guards – while going around the cycle B (discarding the cut vertices), non-cut vertices are alternatelyguarded and unguarded – one B -component G i ∈ T ( G, B ) contains exactly evc x i ( G i ) guards. – all other G j ∈ T ( G, B ) contain exactly evc x j ( G j ) − – each G i ∈ T ( G, B ) contains exactly evc x i ( G i ) guards.When there is an attack on an edge of a Type 1 B -component G j containing only evc x j ( G j ) − B -component G i that presently has evc x i ( G i ) guards will have one less guard and G j gets one moreguard. Since G i and G j are Type 1, this is always possible by an appropriate shifting of guardsthrough the cycle B . In this way, the invariants stated above can be maintained consistently.Hence, evc( G ) is as given in the statement of the lemma.b. when n − k is odd.In this case, note that (cid:100) n − k (cid:101) = (cid:100) n − k +12 (cid:101) . As noted earlier, l is odd and l ≥
1. Using similararguments as in the case when there are no Type 1 B -components, we can show that evc( G ) is atleast (cid:6) n − k (cid:7) + χ ( G, B ). With these many guards, it is possible to protect G keeping the followinginvariants in all configurations. – All cut vertices have guards – at most one G i ∈ T ( G, B ) contains exactly evc x i ( G i ) guards – all other G j ∈ T ( G, B ) contain exactly evc x j ( G j ) − – the maximal uneventful paths P and Q in B that are respectively clockwise and anticlockwisenearest to the Type 1 B -component with evc x i ( G i ) guards (if it exists) respectively contain (cid:98) | V ( P ) | (cid:99) and (cid:98) | V ( Q ) | (cid:99) guards – each G i ∈ T ( G, B ) contains exactly evc x i ( G i ) guards.Hence, in this case also, the lemma holds. 8he following observation gives a method to compute the type of a graph with respect to degree-twovertices in blocks which are cycles. Lemma 6.
Let B be a cycle of n vertices, forming a block of a connected graph G . Let X be the set ofcut vertices of G and let k = | X ∩ V ( B ) | . Suppose each B -component in C B ( G ) satisfies the substructureproperty. Let v ∈ V ( B ) \ X . The type of G with respect to v can be computed as follows. • If T ( G, B ) (cid:54) = ∅ and n − k is even, then evc v ( G ) = evc( G ) = evc( G + v ) . If T ( G, B ) (cid:54) = ∅ and n − k isodd, then evc v ( G ) = evc( G ) + 1 = evc( G + v ) . In both cases, G is Type 1 with respect to v . • If T ( G, B ) = ∅ and n − k is even, then evc v ( G ) = evc( G ) + 1 = evc( G + v ) and G is Type 1 with respectto v . If T ( G, B ) = ∅ and n − k is odd, then evc v ( G ) = evc( G ) < evc( G + v ) = evc v ( G ) + 1 and G isType 2 with respect to v .Proof. This can be proved using Lemma 5 and Lemma 3. Let G (cid:48) be the graph obtained by deleting thetwo edges incident on v from G and adding an edge between its neighbors. Let B (cid:48) be the cycle obtainedby deleting the two edges incident on v from B and adding an edge between its neighbors. We have | X ∩ V ( B ) | = k = | X ∩ V ( B (cid:48) ) | and | V ( B (cid:48) ) | = | V ( B ) | − n −
1. Further, if ˜ X is the set of cut vertices of G + v , then | ˜ X | = | X + 1 | = k + 1. • If T ( G, B ) (cid:54) = ∅ , then using Lemma 5 for G and G (cid:48) , we can see that when n − k is odd, evc( G (cid:48) ) =evc( G ) and when n − k is even, evc( G (cid:48) ) = evc( G ) −
1. Therefore, by Lemma 3, when n − k is odd,evc v ( G ) = evc( G ) + 1 and when n − k is even, evc v ( G ) = evc( G ). Further, using Lemma 5 for G + v ,we can see that when n − k is odd, evc( G + v ) = evc( G ) + 1 and when n − k is even, evc( G + v ) = evc( G ).Thus, in both cases, evc v ( G ) = evc ( G + v ) and hence, G is Type 1 with respect to v . • If T ( G, B ) = ∅ , then using Lemma 5 for G and G (cid:48) , we can see that when n − k is even, evc( G (cid:48) ) =evc( G ) and when n − k is odd, evc( G (cid:48) ) = evc( G ) −
1. Therefore, by Lemma 3, when n − k is even,evc v ( G ) = evc( G )+1 and when n − k is odd, evc v ( G ) = evc( G ). Further, using Lemma 5 for G + v , we cansee that when n − k is even, evc( G + v ) = evc( G ) + 1 and when n − k is odd, then evc( G + v ) = evc( G ) + 1.Therefore, when n − k is even, evc v ( G ) = evc( G + v ) and when n − k is odd, evc( G + v ) = evc v ( G ) + 1.Hence, when n − k is even, G is Type 1 with respect to v and when n − k is odd, G is Type 2 withrespect to v .
4. Computing eternal vertex cover number of cactus graphsTheorem 1.
Every cactus graph satisfies substructure property.Proof.
Let G be a cactus graph. The proof is using an induction on the number of cut-vertices in G .In the base case, G is a cactus without a cut vertex. Then, G is either a single vertex, a single edge or asimple cycle. In all these cases, the lower bound for the number of guards on V ( G ), specified by substructureproperty is equal to the vertex cover number of the respective graphs. Hence, the theorem holds in the basecase.Now, let us assume that the theorem holds for any cactus with at most k cut-vertices. Let G be a cactuswith k + 1 cut vertices, for k ≥ X be the set of cut vertices of G . Let v be a non-cut vertex of G and G (cid:48) be an arbitrary v -extension of G . We need to show that in any eternal vertex cover configuration of G (cid:48) , the number of guards on V ( G ) is as specified by the substructure property. Since v is a non-cut vertexof the cactus G , either it is a degree-one vertex of G or it is a degree-2 vertex of G that is in some block B of G , where B is a cycle. We want to compute a lower bound on the number of guards on V ( G ) in anarbitrary eternal vertex cover configuration C (cid:48) of G (cid:48) .First, consider the case where v is a degree-one vertex of G . Let w be the neighbor of v in G and let H = G \ v . Since G has at least one cut-vertex, w must be a cut vertex in G .9 When w is a cut-vertex of H : Consider any w -component H i of H . H i is a cactus and the numberof its cut vertices is less than that of G . Hence, H i satisfies the substructure property. Further, G (cid:48) is a w -extension of H i . In C (cid:48) , the number of guards on V ( H i ) is at least evc w ( H i ) − H i ∈ T ( H, w ) and it is evc w ( H i ) if H i ∈ T ( H, w ). The total number of guards on V ( H ) is at least1 + (cid:80) H i ∈ T ( H,w ) (evc w ( H i ) −
2) + (cid:80) H i ∈ T ( H,w ) (evc w ( H i ) − v is occupied in C (cid:48) , then the totalnumber of guards on V ( G ) is at least the same as that of evc( G ) as given by Lemma 1. If v is notoccupied in C (cid:48) , then to defend an attack on the edge wv , a guard from V ( H ) must move to v . Hence,in C (cid:48) , the number of guards in V ( H ) should have been one more than the minimum mentioned earlier.Hence, in this case also, the number of guards on V ( G ) in C (cid:48) would have been at least evc( G ).By Lemma 1, evc( G + v ) = evc( G ) + 1. By Lemma 2, the type of G with respect to v is the same as thetype of H with respect to w . Hence, if H is Type 1 with respect to w , we have evc( G + v ) = evc v ( G ) =evc( G ) + 1. We have seen that the number of guards on V ( G ) is at least evc( G ) = evc v ( G ) −
1. If H is Type 2 with respect to w , then we have evc( G + v ) > evc v ( G ). Since evc( G + v ) = evc( G ) + 1, in thiscase we must have evc v ( G ) = evc( G ). We have seen that the number of guards on V ( G ) is at leastevc( G ) = evc v ( G ). Hence, the substructure property holds in both cases. • When w is not a cut vertex of H : In this case, G (cid:48) is a w -extension of H . By substructure property of H , it follows that in configuration C (cid:48) , the number of guards on V ( G ) must be at least evc w ( H ) if H is Type 1 with respect to w and at least evc w ( H ) + 1 if H is Type 2 with respect to w . By Lemma 1,when H is Type 1 with respect to w , evc( G ) = evc w ( H ) and when H is Type 2 with respect to w ,evc( G ) = evc w ( H ) + 1. By Lemma 2, the type of G with respect to v is the same as the type of H withrespect to w . Hence, if H is Type 1 with respect to w , we have evc( G + v ) = evc v ( G ) = evc( G ) + 1. Thenumber of guards on V ( G ) is at least evc w ( H ) = evc v ( G ) −
1. If H is Type 2 with respect to w , wehave evc( G + v ) > evc v ( G ). Since evc( G + v ) = evc( G ) + 1, in this case we must have evc v ( G ) = evc( G ).The number of guards on V ( G ) is at least evc w ( H ) + 1 = evc v ( G ). Hence, the substructure propertyholds in both cases.Now, consider the case when v is a degree-two vertex of G that is in some block B of G , where B is acycle. Suppose | V ( B ) | = n b . Let X be the set of cut vertices of G and k b = | X ∩ V ( B ) | . As noted earlier,we have to compute a lower bound on the number of guards on V ( G ) in an arbitrary eternal vertex coverconfiguration C (cid:48) of G (cid:48) . Let p and q respectively be the clockwise and anticlockwise nearest vertices to v in X ∩ V ( B ). Let P be the path in B between p and q that does not contain v . Note that, every B -componentof G satisfies substructure property by our induction hypothesis and G (cid:48) is an extension for each of them.Hence, we have a lower bound on the number of guards on V ( G i ), for each G i ∈ C B ( G ). Similarly, thecondition stated in Lemma 4 needs to be satisfied for each eventful subpath P (cid:48) of P . • If T ( G, B ) = ∅ : By similar arguments as in the proof of Lemma 5, we can see that the number of guardson V ( G ) in C (cid:48) must be at least evc( G ). By Lemma 6, when n b − k b is even, G is Type 1 with respectto v and evc v ( G ) = evc( G ) + 1. Since the number of guards on V ( G ) is at least evc( G ) = evc v ( G ) − n b − k b is odd, G is Type 2 with respect to v and evc v ( G ) = evc( G ) andwe are done. • If T ( G, B ) (cid:54) = ∅ and n b − k b is even: By similar arguments as in the proof of Lemma 5, we can see thatthe number of guards on V ( G ) in C (cid:48) must be at least evc( G ) −
1. By Lemma 6, G is Type 1 with respectto v and evc( G ) = evc v ( G ). Since the number of guards on V ( G ) is at least evc( G ) − v ( G ) − • T ( G, B ) (cid:54) = ∅ and n b − k b is odd: By similar arguments as in the proof of Lemma 5, we can see thatthe number of guards on V ( G ) in C (cid:48) must be at least evc( G ). By Lemma 6, G is Type 1 with respectto v and evc v ( G ) = evc( G ) + 1. Since the number of guards on V ( G ) is at least evc( G ) = evc v ( G ) − V ( G ) in an arbitrary eternal vertex coverconfiguration C (cid:48) of G (cid:48) satisfies the condition stated in substructure property. Hence, G satisfies substructureproperty. 10hus, by induction, it follows that every cactus satisfies substructure property.Now, we have all ingredients for designing a recursive algorithm for the computation of eternal vertexcover number of a cactus, using Lemma 1. Our algorithm will take a cactus G and a vertex v of G andoutput evc( G ), evc v ( G ) and the type of G with respect to v . If G is a cycle or an edge or a vertex, theanswer is trivial and can be computed in linear time.In other cases, G has at least one cut vertex. If v is a cut vertex, then we call the algorithm recursivelyon each v -component of G along with vertex v . Then, we can use Lemma 1 to compute evc( G ) and evc v ( G )in constant time from the result of the recursive call. Using the same information from recursive calls, thetype of G with respect to v can also be computed using Observation 3. If v is a pendent vertex and w isits neighbor in G , then we recursively call the algorithm on ( G \ v, w ). By Lemma 2, the type of G withrespect to v is the same as the type of G \ v with respect to w . Moreover, from the proof of Lemma 2, wehave evc v ( G ) = evc w ( G \ v ) + 1. Further, evc( G ) = evc w ( G \ v ), when G is Type 1 with respect to v andevc( G ) = evc w ( G \ v ) + 1, when G is Type 2 with respect to v . Thus, from the results of the recursive call on( G \ v, w ), the output can be computed in constant time. In the remaining case, v is a vertex that belongsto a cycle B in G . In this case, we recursively call the algorithm for each B -component of G , along with therespective cut vertices its shares with B . Using this information, we can compute evc( G ) using Lemma 5 intime proportional to the number of B -components. We can also compute evc v ( G ) and the type of G withrespect to v , using Lemma 6 in time proportional to the number of B -components.Thus, the algorithm works in all cases and runs in time linear in the size of G . Hence, we have thefollowing result. Theorem 2.
Eternal vertex cover number of a cactus G can be computed in time linear in the size of G . From the upper bound arguments in the proofs discussed in Section 2 and Section 3, we can see thatwith evc( G ) guards determining configurations of guards to keep defending attacks on G is straightforward.
5. Extension to other graph classes
It may be noticed that most of the intermediate results stated in this paper are generic, though we havestated Lemma 5 and Lemma 6 in a way suitable for handling cactus graphs. In this section, we show howto extend the method used for cactus graphs to a graph class which is somewhat more general. We considerconnected graphs in which each block is a cycle, an edge or a biconnected chordal graph.To generalize the proof of Theorem 1 for this class, the base case of the proof needs to be modified tohandle biconnected chordal graphs as well. The following observation addresses this requirement.
Observation 5.
Every biconnected chordal graph satisfies substructure property.Proof.
Let G be a biconnected chordal graph and v ∈ V ( G ). Let G (cid:48) be a v -extension of G and C be aneternal vertex cover configuration of G (cid:48) . If the number of guards on V ( G ) in C is less than mvc v ( G ), then v is not occupied in C . In this configuration, an attack on an edge of G adjacent to v cannot be defended,because when a guard on a vertex of G moves to v , some edge of G will be without guards. Therefore, inany arbitrary eternal vertex cover configuration C of G (cid:48) , the number of guards on V ( G ) must be at leastmvc v ( G ).By a result in [14], evc v ( G ) ∈ { mvc v ( G ) , mvc v ( G ) + 1 } and evc( G + v ) = mvc( G + v ) + 1 = mvc v ( G ) + 1. Ifevc v ( G ) = mvc v ( G ) and evc( G + v ) = mvc v ( G ) + 1, then G is Type 2 with respect to v and we need at leastevc v ( G ) = mvc v ( G ) guards on V ( G ). If evc v ( G ) = mvc v ( G ) + 1 = evc( G + v ), then G is Type 1 with respectto v and we need at least evc v ( G ) − v ( G ) guards on V ( G ). Thus, in both cases, the requirementsof substructure property are satisfied by G .The following lemma is a suitable modification of Lemma 5 and Lemma 6 to handle the new class.11 emma 7. Let B be a biconnected chordal graph forming a block of a connected graph G and let X bethe set of cut vertices of G . Suppose each B -component G i of G that belongs to C B ( G ) satisfies the sub-structure property. If T ( G, B ) = ∅ , then evc( G ) = evc X ∩ V ( B ) ( B ) + χ ( G, B ) − | X ∩ V ( B ) | and evc( G ) =mvc X ∩ V ( B ) ( B ) + 1 + χ ( G, B ) − | X ∩ V ( B ) | otherwise. Further, if χ ( G, B ) is known, then for any v ∈ V ( B ) ,the type of G with respect to v can be computed in time quadratic in the size of B .Proof. Since B a biconnected chordal graph, if for every v ∈ V ( B ) \ X , mvc ( X ∩ V ( B )) ∪{ v } ( B ) = mvc X ∩ V ( B ) ( B ),then evc X ∩ V ( B ) ( B ) = mvc X ∩ V ( B ) ( B ) and evc X ∩ V ( B ) ( B ) = mvc X ∩ V ( B ) ( B ) + 1 otherwise [14]. Consider anyeternal vertex cover class C of G . By substructure property of B -components of G , it follows that in any con-figuration of C , the number of guards on (cid:83) G i ∈C B V ( G i ) is at least χ ( G, B ). To cover edges of the induced sub-graph of G on V ( B ) \ X , the number of guards required is at least mvc( B \ X ) = mvc X ∩ V ( B ) ( B ) −| X ∩ V ( B ) | .Hence, the total number of guards on V ( G ) must be at least mvc X ∩ V ( B ) ( B ) − | X ∩ V ( B ) | + χ ( G, B ).Further, if there is a vertex v ∈ V ( B ) \ X for which mvc ( X ∩ V ( B )) ∪{ v } ( B ) (cid:54) = mvc X ∩ V ( B ) ( B ), then inany configuration of C in which v is occupied, the total number of guards on V ( G ) must be at leastmvc X ∩ V ( B ) ( B )+1 −| X ∩ V ( B ) | + χ ( G, B ). Hence, in all cases, evc( G ) ≥ evc X ∩ V ( B ) ( B )+ χ ( G, B ) −| X ∩ V ( B ) | .a. When T ( G, B ) = ∅ : In this case, it is easy to show that evc( G ) ≤ evc X ∩ V ( B ) ( B )+ χ ( G, B ) −| X ∩ V ( B ) | .b. When T ( G, B ) (cid:54) = ∅ : In this case, by repeated attacks on edges of V ( G i ) for some G i ∈ T ( G, B ),eventually a configuration which requires χ ( G, B ) + 1 guards on (cid:83) G i ∈C B V ( G i ) can be forced. Hence,evc( G ) ≥ mvc X ∩ V ( B ) ( B ) + 1 + χ ( G, B ) − | X ∩ V ( B ) | . Since evc X ∩ V ( B ) ( B ) ≤ mvc X ∩ V ( B ) ( B ) + 1, itis not difficult to also show that evc( G ) ≤ mvc X ∩ V ( B ) ( B ) + 1 + χ ( G, B ) − | X ∩ V ( B ) | .In both cases, the value of evc( G ) is as stated in the lemma. For deciding the type of G with respect toa vertex v ∈ V ( B ), we need to compare evc v ( G ) and evc( G + v ). For computing evc( G + v ), we can use theformula given by the first part of the lemma for the graph evc( G + v ). Using similar arguments as in the proofof the first part of the lemma, we get the following: if T ( G, B ) = ∅ , then evc v ( G ) = evc ( X ∩ V ( B )) ∪{ v } ( B ) + χ ( G, B ) − | X ∩ V ( B ) | and evc v ( G ) = mvc ( X ∩ V ( B )) ∪{ v } ( B ) + 1 + χ ( G, B ) − | X ∩ V ( B ) | otherwise. Bycomputing mvc ( X ∩ V ( B )) ( B ) and mvc ( X ∩ V ( B )) ∪{ v } ( B ) for each v ∈ V ( B ), evc v ( G ) and evc( G + v ) can becomputed. Since minimum vertex cover of a chordal graph can be computed in linear time, the total timerequired for computing evc v ( G ) and evc( G + v ) this way is possible in time quadratic in the size of B . Fromthe values of evc v ( G ) and evc( G + v ), the type of G with respect to v can be inferred.Now, similar arguments as in the proof of Theorem 1 and Theorem 2 yields: Theorem 3.
Suppose G is a connected graph in which each block is a cycle, an edge or a biconnected chordalgraph. Then, G satisfies substructure property and the eternal vertex cover number of G can be computed intime quadratic in the size of G .
6. A cactus for which other lower bounds are weak
It is known that if G is a graph with X being the set of its cut vertices, and mvc X ( G ) is the minimumcardinality of a vertex cover of G that contains all vertices of X , then evc( G ) ≥ mvc X ( G ) [3]. For cycles,this lower bound is equal to the eternal vertex cover number and for trees the difference of the eternal vertexcover number and this lower bound is at most one. But, it is interesting to note that the lower boundcould be much smaller than the optimum for some cactus graphs. An example of this is shown in Figure 1.Using the formula given by Lemma 5, we can show that the eternal vertex cover number of this graph G is k + (cid:100) k +12 (cid:101) . However, for this graph, mvc X ( G ) is only k making evc( G ) > . X ( G ).
7. Discussion and Open problems
The lower bounding method based on the substructure property of cactus graphs presented here showsthat the method could be potentially useful in obtaining efficient algorithms for computing the eternal12 v v v v k v v k − Figure 1: A cactus G in which pendent vertices are attached to alternate vertices of an even cycle on 2 k vertices. The cactus G has a minimum vertex cover of size k containing all cut vertices of G and evc( G ) = k + (cid:100) k +12 (cid:101) . vertex cover number of graphs that belong to classes for which other known lower bound techniques are noteffective. Though the substructure property has been applied here in context of cactus graphs, we believethat this property or a generalization of it holds for fairly large classes of graphs (if not all graphs). We listbelow a few questions that appear interesting in this context. • Do all graphs satisfy the substructure property? If not, can we characterize graphs that satisfy thisproperty? • Is it possible to generalize substructure property and use it for the computation of eternal vertex covernumber of outerplanar graphs and bounded treewidth graphs? • In the restricted version of the eternal vertex cover problem in which at most one guard is permittedon a vertex in any configuration, is it true in general that for any graph G and vertex x of G ,evc( G ) ≤ evc ( G x ) + ?The last question looks deceptively simple (and is trivial if multiple guards are allowed on a vertex in aconfiguration). For graphs that satisfy substructure property, the answer to this question is affirmative (byRemark 2 and Observation 3). But, we do not know the answer in general.
8. Acknowledgments
We would like to thank Ilan Newman, University of Haifa, for introducing the problem and for his usefulsuggestions.
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