Polynomial time recognition of vertices contained in all (or no) maximum dissociation sets of a tree
aa r X i v : . [ m a t h . C O ] F e b Polynomial time recognition of vertices contained in all (or no) maximumdissociation sets of a tree
Jianhua Tu a,b,c , Lei Zhang a , Junfeng Du a, ∗ a Department of mathematics, Beijing University of Chemical Technology,Beijing, P.R. China 100029 b Key Laboratory of Tibetan Information Processing and Machine Translation,Qinghai Province, XiNing, P.R. China 810008 c Key Laboratory of Tibetan Information Processing, Ministry of Education,XiNing, P.R. China, 810008
Abstract
Given a graph G , a subset of vertices is called a maximum dissociation set ifit induces a subgraph of maximum degree at most 1 and the subset has maximumcardinality. In this paper, we first characterize the set of vertices of a tree that arecontained in all, or in no, maximum dissociation sets of the tree. Then we presenta linear time recognition algorithm which can determine whether a given vertex ina tree is in all (or no) maximum dissociation sets of the tree. Thus we can find allvertices contained in all (or no) maximum dissociation sets of a tree of order n in O ( n ) time. Keywords:
Maximum dissociation set; Dissociation number; Tree; Polynomialtime algorithm
The maximum independent set problem is a widely studied classical problem of graphtheory and has important applications in many fields. Different from the classical ex-tremum and extremal graph characterization, in 1982, Hammer et al. [7] offered a ideato study this problem, and investigated the set of vertices that are contained in all orin no maximum independent sets of a graph. Then, this kind of problem for some othervertex subsets with given properties has been studied by many researchers. For example,Mynhardt [10], Cockayne et al. [6], and Blidia et al. [2] also have considered this kind ofproblem for minimum dominating sets, total dominating sets, and minimum double dom-inating sets of trees, respectively. As a natural extension of independent set, dissociationset has become an important and interesting topic. Therefore, the focus of the presentpaper is on the characterization of the set of vertices that are contained in all, or in no,maximum dissociation sets of trees. ∗ Corresponding author.E-mail addresses: [email protected] (J. Tu), [email protected] (L. Zhang),[email protected] (J. Du) e consider only finite, simple, and undirected labeled graphs, and use Bondy andMurty [3] for terminology and notations not defined here.Let G = ( V ( G ) , E ( G )) be a graph and let v be a vertex in G . The (open) neighborhoodof v , denoted by N G ( v ), is the set of vertices that are adjacent to v in G . The closedneighborhood of v , denoted by N G [ v ], is the set of vertices N G ( v ) ∪ { v } . The degree of v ,denoted by d G ( v ), is the cardinality of N G ( v ). Let U be a subset of V ( G ). The subgraphof G induced by U is denoted by G [ U ]. The subgraph G [ V ( G ) \ U ] is denoted by G − U .Furthermore, G − U can be written by G − u if U = { u } .A dissociation set in a graph G is a set of vertices F such that the subgraph G [ F ]has maximum degree at most 1, i.e., G [ F ] consists of isolated vertices and isolated egdes.A maximum dissociation set of G is a dissociation set of maximum cardinality. The dissociation number of a graph G , denoted by ψ ( G ), is the cardinality of a maximumdissociation set of G . Define the vertex subsets A ( G ), F ( G ) and N ( G ) by A ( G ) = { v ∈ V ( G ) : v is in all maximum dissociation sets of G } , F ( G ) = { v ∈ V ( G ) : v is in some but not all maximum dissociation sets of G } , N ( G ) = { v ∈ V ( G ) : v is in no maximum dissociation set of G } . The concept of dissociation set was introduced by Yannakakis [15] in 1981. The max-imum dissociation set problem, i.e., the problem of finding a dissociation set of maximumsize in a given graph is known to be NP-hard for bipartite graphs [15]. The complexityof this problem on some other graphs also has been studied in [1, 4, 5, 11]. A k -path in G is a not necessarily induced path of order k . Note that a set F of vertices of a graph G is a dissociation set if and only if its complement V ( G ) \ F is a 3-path vertex cover,that is, a set of vertices of G intersecting every 3-path in G . The 3-path vertex coverproblem, i.e., the problem of finding a minimum 3-path vertex cover in a graph G hasreceived considerable attention in the literature [4, 8, 9, 13, 14].In this paper, we investigate and characterize the vertex subsets A ( T ) and N ( T ) ofa tree T . The study is inspired by the relationship between the characteristic of vertexsubsets A ( G ) and N ( G ) and the number of maximum dissociation sets in a graph G .In [12], Tu, Zhang and Shi found four structure theorems concerning the vertex subsets A ( G ) and N ( G ) and determined the maximum number of maximum dissociation sets ina tree of order n .The paper is organized as follows. In next section, some necessary notations, andlemmas will be introduced. In section 3, the vertex subsets A ( T ) and N ( T ) of a tree T will be characterized. In section 4, a linear recognition algorithm which can determinewhether a given vertex in a tree is in all (or no) maximum dissociation sets of the treewill be presented. Thus, using the recognition algorithm, all vertices contained in all (oro) maximum dissociation sets of a tree of order n can be found in O ( n ) time. A rooted tree T is a tree with a specified vertex r , called the root of T . An orientationof a rooted tree in which every vertex but the root has indegree one. Let T be a treewith root r , then each vertex on the path rT v , including the vertex v itself, is called an ancestor of v , and each vertex of which v is an ancestor is a descendant of v . An ancestor or descendant of a vertex is proper if it is not the vertex itself. The immediate properancestor of a vertex v other than the root is its parent , denoted by p ( v ), and the verticeswhose parent is v are its children . Define the vertex subsets C T ( v ), D T ( v ) and D T [ v ] by C T ( v ) = { u ∈ V ( T ) : u is a child of v } ,D T ( v ) = { u ∈ V ( T ) : u is a proper descendant of v } ,D T [ v ] = D ( v ) ∪ { v } . If no confusion occurs, these also be written by C ( v ), D ( v ) and D [ v ], respectively. Thesubtree induced by D T [ v ] is denoted by T v .A vertex in a tree T is called a leaf if it has degree 1 and a branch vertex if it hasdegree at least 3. The set of branch vertices of T is denoted by B ( T ). A path P in T issaid to be a v − L path, if P joins v to a leaf of T . Denote the order of P by n ( P ), andfor i = 0 , ,
2, define C i ( v ) = { u ∈ C ( v ) : T u contains a u − L path P with n ( P ) ≡ i mod 3 } Now, we give some basic observations about maximum dissociation sets of the path.
Observation 2.1.
Let P n be a path of order n with n ≥ and u, v be the leaves of P n .(a) ψ ( P n ) = n + i , where n ≡ i (mod3) , i = 0 , , .(b) If n ≡ , then P n has a maximum dissociation set that contains exactlyone leaf.(c) If n ≡
1( mod 3) , then every maximum dissociation set of P n contains both leaves of P n , furthermore, there exists a maximum dissociation set F in P n such that d P n [ F ] ( u ) = 0 .(d) If n ≡ , then P n has only one maximum dissociation set F , furthermore, { u, v } ⊂ F and d P n [ F ] ( u ) = d P n [ F ] ( v ) = 1 . Our next purpose is to characterize A ( T ) and N ( T ) in the case where B ( T ) ≤ emma 2.2. Let T be a tree rooted at v such that d T ( u ) ≤ for all u ∈ V ( T ) − { v } .Then ψ ( T ) = P w ∈ C ( v ) ψ ( T w ) + 1 , if | C ( v ) | = 0 and | C ( v ) | ≤ ; P w ∈ C ( v ) ψ ( T w ) , otherwise. Proof.
Since T w is a path for each w ∈ C ( v ), ψ ( T w ) is easy to determine and C i ( v ) ∩ C j ( v ) = ∅ for i = j . Note that P w ∈ C ( v ) ψ ( T w ) ≤ ψ ( T ) ≤ P w ∈ C ( v ) ψ ( T w ) + 1. We considerthe two cases. Case 1. | C ( v ) | = 0 and | C ( v ) | ≤ w ∈ C ( v ), then T w ∼ = P n with n ≡ F w be a maximum dissociationset of T w such that w / ∈ F w ( F w exists by Observation 2.1(b)). If w ∈ C ( v ), then T w ∼ = P n with n ≡ F w be a maximum dissociation set of T w such that d T w [ F w ] ( w ) = 0 ( F w exists by Observation 2.1(c)). Now, let F = [ w ∈ C ( v ) F w ∪ { v } , (1)then F is a dissociation set of T and | F | = P w ∈ C ( v ) ψ ( T w ) + 1. Thus, F is a maximumdissociation set of T and ψ ( T ) = P w ∈ C ( v ) ψ ( T w ) + 1. Case 2. | C ( v ) | ≥ | C ( v ) | ≥ ψ ( T ) = P w ∈ C ( v ) ψ ( T w ) + 1. Let F be a maximumdissociation set of T . Then, v ∈ F and F ∩ T w is a maximum dissociation set of T w foreach w ∈ C ( v ). Let F w := F ∩ T w for each w ∈ C ( v ). If w ∈ C ( v ), then T w ∼ = P n with n ≡ w ∈ F w and d T w [ F w ] ( w ) = 1. Thus, if | C ( v ) | ≥ T [ F ] that contains the vertex v , a contradiction.If w ∈ C ( v ), then T w ∼ = P n with n ≡ w ∈ F w . Thus, if | C ( v ) | ≥
2, then there is a 3-path in T [ F ] that contains the vertex v ,a contradiction.The proof is complete. Theorem 2.3.
Let T be a tree rooted at v such that d T ( u ) ≤ for all u ∈ V ( T ) − { v } .Then(a) v ∈ A ( T ) if and only if | C ( v ) | = 0 and | C ( v ) | ≤ ;(b) v ∈ N ( T ) if and only if | C ( v ) | = 2 or | C ( v ) | + | C ( v ) | ≥ . Proof. (a) Necessity. Suppose, for a contradiction, that | C ( v ) | ≥ | C ( v ) | ≥
2. Let F = S w ∈ C ( v ) F w , where F w is a maximum dissociation set of T w for each w ∈ C ( v ). Byemma 2.2, we have ψ ( T ) = P w ∈ C ( v ) ψ ( T w ). Thus, F is a maximum dissociation set of T and v / ∈ F , which contradicts with v ∈ A ( T ).Sufficiency. Suppose that | C ( v ) | = 0 and | C ( v ) | ≤
1. Then ψ ( T ) = P w ∈ C ( v ) ψ ( T w )+1 by Lemma 2.2 and the vertex v is in all maximum dissociation sets of T . Thus, v ∈ A ( T ).The proof of (a) is complete.(b) Necessity. Suppose, for a contradiction, that | C ( v ) | 6 = 2 and | C ( v ) | + | C ( v ) | ≤ | C ( v ) | = 0 and | C ( v ) | ≤
1, then v ∈ A ( T ) by (a), a contradiction.If | C ( v ) | = 0 and | C ( v ) | = 2, then we assume w , w ∈ C ( v ). For each w ∈ C ( v ),there exists a maximum dissociation set F w of T w such that w / ∈ F w by Observation2.1(b). For w ∈ C ( v ), we have T w − w ∼ = P n with n ≡ F w be amaximum dissociation set of T w − w . Then | F w | = ψ ( T w ) − w ∈ C ( v ), there exists a maximum dissociation set F w of T w such that d T w [ F w ] ( w ) = 0by Observation 2.1(c). Let F = S w ∈ C ( v ) F w ∪ F w ∪ F w ∪ { v } , then F is a dissociationset of T and | F | = P w ∈ C ( v ) ψ ( T w ). By Lemma 2.2, F is a maximum dissociation set of T and v ∈ F , a contradiction.If | C ( v ) | = 1 and | C ( v ) | ≤
1, then we assume w ∈ C ( v ). For w ∈ C ( v ), we have T w − w ∼ = P n with n ≡ F w be a maximum dissociation set of T w − w ,then | F w | = ψ ( T w ) − w ∈ C ( v ), let F w be a maximumdissociation set of T w such that w / ∈ F w . For w ∈ C ( v ), let F w be a maximum dissociationset of T w such that d T w [ F w ] ( w ) = 0. Now let F = S w ∈ C ( v ) −{ w } F w ∪ F w ∪ { v } , then F is a dissociation set of T and | F | = P w ∈ C ( v ) ψ ( T w ). By Lemma 2.2, F is a maximumdissociation set of T and v ∈ F , a contradiction.Sufficiency. Suppose for a contradiction that F is a maximum dissociation set of T and v ∈ F . For each w ∈ C ( v ), | F ∩ T w | ≤ ψ ( T w ). If w ∈ C ( v ), then T w ∼ = P n with n ≡ v ∈ F , we have | F ∩ T w | ≤ ψ ( T w ) −
1. If w ∈ C ( v ), then everymaximum dissociation set F w of T w contains the vertex w . Since v ∈ F , there are at least | C ( v ) | − w in C ( v ) such that | F ∩ T w | ≤ ψ ( T w ) −
1. Hence, it is easy to check | F | < P w ∈ C ( v ) ψ ( T w ) ≤ ψ ( T ), a contradiction.The proof of (b) is complete. A ( T ) and N ( T ) We describe the pruning process , which was introduced in [10] and will allow us to useTheorem 2.3 to characterize A ( T ) and N ( T ) for an arbitrary tree T .et T be a rooted tree with the root v . Let u be a branch vertex at maximum distancefrom v . Note that | C ( u ) | ≥ d T ( x ) ≤ x ∈ D ( u ). If u = v , we apply thefollowing pruning process: • if | C ( u ) | ≥ | C ( u ) | ≥
2, then delete D [ u ], • if | C ( u ) | = 0 and | C ( u ) | ≤
1, then for all w ∈ C ( u ) − { z } , delete D [ w ], where z isthe vertex in C ( u ) if | C ( u ) | = 1 , otherwise z is any one vertex in C ( u ).This step of pruning process is called a pruning of T at u . Repeat the above pruningprocess until a tree ¯ T is obtained with d ¯ T ( u ) ≤ u ∈ V ( ¯ T ) − { v } . The tree¯ T is unique and is called the pruning of T . We now show that the vertex v is in allmaximum dissociation sets (or in no maximum dissociation set) of T if and only if v is inall maximum dissociation sets (or in no maximum dissociation set) of the pruning ¯ T of T . Lemma 3.1.
Let T be a rooted tree with the root v and ¯ T be the pruning of T . For everymaximum dissociation set ¯ F of ¯ T , there exists a maximum dissociation set F of T suchthat v ∈ F if and only if v ∈ ¯ F . Conversely, for every maximum dissociation set F of T ,there exists a maximum dissociation set ¯ F of ¯ T such that v ∈ ¯ F if and only if v ∈ F . Proof.
We prove the lemma by induction on | B ′ ( T ) | , where B ′ ( T ) = { u ∈ V ( T ) \ { v } : d ( u ) ≥ } . If | B ′ ( T ) | = 0, then T = ¯ T and the result follows trivially. Suppose that thelemma holds when | B ′ ( T ) | < k , and let T be a tree with | B ′ ( T ) | = k . Let u be a vertexof B ′ ( T ) at maximum distance from v . Let T ′ be the tree obtained from T by applyinga pruning of T at u . Thus, ¯ T is also the pruning of T ′ .First, we show that for every maximum dissociation set ¯ F of ¯ T , there exists a maximumdissociation set F of T such that v ∈ F if and only if v ∈ ¯ F . By the induction hypothesis,for every maximum dissociation set ¯ F of ¯ T , there exists a maximum dissociation set F ′ of T ′ such that v ∈ F ′ if and only if v ∈ ¯ F .We consider the following two cases. Case 1. | C ( u ) | = 0 and | C ( u ) | ≤ w ∈ C ( u ) − { z } , T w ∼ = P n with n ≡ F w be a maximumdissociation set of T w such that w / ∈ F w ( F w exists by Observation 2.1(b)). Let F = S w ∈ C ( u ) −{ z } F w ∪ F ′ . It is easy to see that F is a maximum dissociation set of T . Since v ∈ F ′ if and only if v ∈ F , we have v ∈ F if and only if v ∈ ¯ F . Case 2. | C ( u ) | ≥ | C ( u ) | ≥ T ′ = T − D [ u ]. For each w ∈ C ( u ), let F w be a maximum dissociationset of T w . Let F = S w ∈ C ( u ) F w ∪ F ′ . Since | C ( u ) | ≥ | C ( u ) | ≥
2, by Lemma 2.2, ψ ( T u ) = P w ∈ C ( u ) ψ ( T w ) = P w ∈ C ( u ) | F w | , which implies that S w ∈ C ( u ) F w is a maximumdissociation set of T u . Thus, F is a maximum dissociation set of T . Since v ∈ F ′ if andonly if v ∈ F , we have v ∈ F if and only if v ∈ ¯ F .Now, we show that for every maximum dissociation set F of T , there exists a maximumdissociation set ¯ F of ¯ T such that v ∈ ¯ F if and only if v ∈ F . Likewise, we consider thefollowing two cases. Case 1. | C ( u ) | = 0 and | C ( u ) | ≤ w ∈ C ( u ) − z , we have T w ∼ = P n with n ≡ F w be a maximumdissociation set of T w such that w / ∈ F w ( F w exists by Observation 2.1(b)).For every maximum dissociation set F of T , let F ′ = F − S w ∈ C ( u ) −{ z } D [ w ]. Then F ′ is a dissociation set of T ′ and v ∈ F if and only if v ∈ F ′ . We will prove that F ′ isa maximum dissociation set of T ′ . Suppose, for a contradiction, that F is a maximumdissociation set of T ′ with | F | > | F ′ | . Let F = S w ∈ C ( u ) −{ z } F w ∪ F . Then F is adissociation set of T and | F | = | F | + X w ∈ C ( u ) −{ z } | F w | > | F ′ | + X w ∈ C ( u ) −{ z } | F ∩ D [ w ] | = | F | , (2)which contradicts with the fact that F is a maximum dissociation set of T . Thus F ′ isa maximum dissociation set of T ′ . By the induction hypothesis, there exists a maximumdissociation set ¯ F of ¯ T such that v ∈ F ′ if and only if v ∈ ¯ F . Hence, there exists amaximum dissociation set ¯ F of ¯ T such that v ∈ F if and only if v ∈ ¯ F . Case 2. | C ( u ) | ≥ | C ( u ) | ≥ w ∈ C ( u ), let F w be a maximum dissociation set of T w . By Lemma 2.2, ψ ( T u ) = P w ∈ C ( u ) ψ ( T w ) = P w ∈ C ( u ) | F w | .For every maximum dissociation set F of T , let F ′ = F − D [ u ]. We will prove that F ′ is a maximum dissociation set of T ′ . Suppose, for a contradiction, that F is a maximumdissociation set of T ′ with | F | > | F ′ | . Let F = S w ∈ C ( u ) F w ∪ F . Then F is a dissociationset of T and | F | = | F | + X w ∈ C ( u ) | F w | > | F ′ | + | F ∩ D [ u ] | = | F | , (3)which contradicts with the fact that F is a maximum dissociation set of T . Thus F ′ isa maximum dissociation set of T ′ . By the induction hypothesis, there exists a maximumdissociation set ¯ F of ¯ T such that v ∈ F ′ if and only if v ∈ ¯ F . Thus, there exists amaximum dissociation set ¯ F of ¯ T such that v ∈ F if and only if v ∈ ¯ F .e complete the proof.By Lemma 3.1, we can obtain the following corollary. Corollary 3.2.
Let T be a rooted tree with the root v and ¯ T be the pruning of T , then v ∈ A ( T ) (or N ( T ) ) if and only if v ∈ A ( ¯ T ) (or N ( ¯ T )) . By Theorem 2.3 and Corollary 3.2, the characterizations of A ( T ) and A ( T ) can beobtained immediately. Theorem 3.3.
Let T be a tree and v be a vertex of T . Let T v be the rooted tree obtainedfrom T with the root v and ¯ T v be the pruning of T v . Then(a) v ∈ A ( T ) if and only if | C T v ( v ) | = 0 and | C T v ( v ) | ≤ ;(b) v ∈ N ( T ) if and only if | C T v ( v ) | = 2 or | C T v ( v ) | + | C T v ( v ) | ≥ . In this section, we present a linear time recognition algorithm which can determinewhether a given vertex in a tree is in all (or no) maximum dissociation sets of the tree.The recognition algorithm is described in detail as follows.For a rooted tree, we can use the breadth-first search algorithm to find the distancefrom the root to each other vertex in linear time. Step 4 is the pruning process of the rootedtree T and can be executed in linear time. Thus, the runtime of the recognition algorithmis linear. Thus, using the recognition algorithm we can find all vertices contained in all(or no) maximum dissociation sets of a tree of order n in O ( n ) time. ecognition AlgorithmInput : a tree T and a vertex v ∈ T . Output : v ∈ A ( T ); or v ∈ F ( T ); or v ∈ N ( T ).1. change the tree T into a rooted tree by choosing the vertex v as the root2. compute the distance d ( v, u ) from v to each other vertex u
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