GENERALIZED INDEPENDENCE IN GRAPHS HAVING CUT-VERTICES

GENERALIZED INDEPENDENCE IN GRAPHS HAVING CUT-VERTICES Vladimir D. Samodivkin 7th January 2008 (Dedicated to Mihail Konstantinov on his 60th birthday) Abstract For a graphical property P and a graph G, a subset S of vertices of G is P-independent if the subgraph induced by S has the property P. The lower P-independence number, i P (G), is the smallest number of vertices in a maximal P-independent set in G. The upper P-independence number, β P (G), is the maximum cardinality of a P-independent set in G. In this paper we present results on β P (G) and i P (G) for graphs having cut-vertices. 2000 MSC: Primary 05C69 Key words: independent set; good/bad/fixed/free vertex; nondegenerate/ induced-hereditary/additive/ graph-property. 1 Introduction All graphs considered in this article are finite, undirected, without loops or multiple edges. For the graph theory terminology not presented here, we follow Haynes et al. [6]. We denote the vertex set and the edge set of a graph G by V (G) and E(G), respectively. The subgraph induced by S V (G) is denoted by S, G. Let G denote the set of all mutually nonisomorphic graphs. A graph property is any non-empty subset of G. We say that a graph G has property P whenever there exists a graph H P which is isomorphic to G. For example we list some graph properties: Department of Mathematics, UACEG, Hristo Smirnenski 1 Blv. vlsam_fte@uacg.bg 1046 Sofia, Bulgaria,

I = {H G : H is totally disconnected}; F = {H G : H is a forest}; UK s = {H G : each component of H is complete and has order at most s}, s 1. A graph property P is called induced-hereditary, if from the fact, that a graph G has the property P, it follows that all induced subgraphs of G also belongs to P. A property is called additive if it is closed under taking disjoint union of graphs. A property P is called nondegenerate if I P ([5]). Note that I, F and UK s are nondegenerate, additive and induced-hereditary properties. For a survey see [1]. Any set S V (G) such that the subgraph S, G satisfies property P is called a P-independent set. A P-independent set S V (G) is a maximal P-independent set if no proper superset S S is P-independent. Lemma 1.1. ([6], Chapter 3, Proposition 3.1) Let G be a graph and let P be an induced-hereditary property. Then a set S V (G) is a maximal P-independent set if and only if for every vertex v V (G) S, S {v} is not P-independent. The set of all maximal P-independent sets of a graph G will be denoted by M P S(G). The lower P-independence number, i P (G) is the smallest number of vertices in a maximal P-independent set in G and the upper P-independence number, β P (G) is the maximum cardinality of an P-independent set in G. If P is nondegenerate then i P (G) and β P (G) always exist and i P (G) β P (G) [5]. Results on i P (G) and β P (G) may be found in [2, 3, 5, 8]. Note that: (a) (b) in the case P = I, every element of M I S(G) is an independent dominating set and the numbers i I (G) and β I (G) are the well known as the independent domination number i(g) and the independence number β 0 (G); in the case P = F, every element of M F S(G) is a maximal acyclic set of G and i F (G) (β F (G)) is denoted by i a (G) (β a (G)) and is called the acyclic (upper acyclic) number. The concept of acyclic numbers of graphs was introduced by Hedetniemi, Hedetniemi and Rall [7]. Let µ(g) be a numerical invariant of a graph G defined in such a way that it is the minimum or maximum number of vertices of a set S V (G) with a given property P. A set with property P and with µ(g) vertices in G is called a µ-set of G. A vertex v of a graph G is defined to be

(a) [4] µ-good if v belongs to some µ-set of G; (b) [4] µ-bad if v belongs to no µ - set of G; (c) [9] µ-fixed if v belongs to every µ-set; (d) [9] µ-free if v belongs to some µ-set but not to all µ-sets. For a graph G, a nondegenerate property P G and µ {i P, β P } we define: G(G, µ) = {x V (G) : x is µ-good }; B(G, µ) = {x V (G) : x is µ-bad }; Fi(G, µ) = {x V (G) : x is µ-fixed }; Fr(G, µ) = {x V (G) : x is µ-free }; V 0 (G, µ) = {x V (G) : µ(g x) = µ(g)}; V (G, µ) = {x V (G) : µ(g x) < µ(g)}; V + (G, µ) = {x V (G) : µ(g x) > µ(g)}. Clearly {G(G, µ), B(G, µ)} and {V (G, µ), V 0 (G, µ), V + (G, µ)} are partitions of V (G), and {Fi(G, µ), Fr(G, µ)} is a partition of G(G). Definition 1.2. We say that a property P G is 1-additive if it is closed under taking union of two graphs having at most one vertex in common. Note that I and F are 1-additive. Let G 1 and G 2 be connected graphs, both of order at least two, and let they have an unique vertex in common, say x. Then a coalescence G 1 x G2 is the graph G 1 G 2. Clearly, x is a cut-vertex of G 1 x G2. In this paper we present results on β P (G) and i P (G) when G is a coalescence of graphs. 2 Upper P-independence number Theorem 2.1. Let G be a nontrivial graph and H G be nondegenerate and induced-hereditary. Then: (1) V (G) = V (G, β H ) V 0 (G, β H ); (2) V (G, β H ) = {x V (G) : β H (G x) = β H (G) 1} = Fi(G, β H ). Proof. (1): If v V (G) and M is a β H -set of G v then M is an H-independent set of G which implies β H (G v) β H (G). (2): Let v V (G) and let M 1 be a β H -set of G. First assume v is not β H -fixed. Hence the set M 1 may be chosen so that v M 1 and then M 1 is an H-independent set of G v implying β H (G) = M 1 β H (G v). Now by (1) it follows β H (G) = β H (G v).

Let v be β H -fixed. Then each β H -set of G v is an H-independent set of G but is not β H -set of G. Hence β H (G) > β H (G v). Since H is induced-hereditary, M 1 {v} is an H-independent set of G v which implies β H (G v) M 1 {v} = β H (G) 1. Theorem 2.2. Let G = G 1 x G2 and let H G be additive and inducedhereditary. Then, (i) β H (G 1 ) + β H (G 2 ) 2 β H (G) β H (G 1 ) + β H (G 2 ); (ii) β H (G) = β H (G 1 ) + β H (G 2 ) if and only if x is no β H -fixed vertex of G i, i = 1, 2; (iii) β H (G) = β H (G 1 ) + β H (G 2 ) 2 if and only if x is a β H -fixed vertex of G i, i = 1, 2 and x is no β H -fixed vertex of G; (iv) if H is 1-additive then β H (G) β H (G 1 ) + β H (G 2 ) 1. Proof. Since H is additive and induced-hereditary, H is nondegenerate. We need the following claims: Claim 1. If x is a β H -fixed vertex of G then β H (G) β H (G 1 ) + β H (G 2 ) 1. If x is no β H -fixed vertex of G then β H (G) β H (G 1 ) + β H (G 2 ). Proof. Let M be a β H -set of G. Since H is induced-hereditary then M i = M V (G i ) is an H-independent set of G i, i = 1, 2. If x M then β H (G) = M = M 1 + M 2 1 β H (G 1 ) + β H (G 2 ) 1. If x M then β H (G) = M = M 1 + M 2 β H (G 1 ) + β H (G 2 ). Claim 2. If x is no β H -fixed vertex of G i, i = 1, 2 then β H (G) β H (G 1 ) + β H (G 2 ). Proof. Let M i be a β H -set of G i and x M i, i = 1, 2. Since H is additive then M = M 1 M 2 is an H-independent set of G and β H (G) M = M 1 + M 2 = β H (G 1 ) + β H (G 2 ). Claim 3. If x is a β H -fixed vertex of G 1 and x belongs to not all β H -sets of G 2 then β H (G) β H (G 1 ) + β H (G 2 ) 1. Proof. Let M j be a β H -set of G j, j = 1, 2 and let x M 2. Since H is additive and induced-hereditary, the set M = (M 1 {x}) M 2 is an H-independent set of G and β H (G) M = M 1 1 + M 2 = β H (G 1 ) + β H (G 2 ) 1.

Claim 4. If x is a β H -fixed vertex of G i, i = 1, 2, then β H (G) β H (G 1 ) + β H (G 2 ) 2. Proof. Since H is additive and induced-hereditary, M = (M 1 M 2 ) {x} is an H- independent set of G providedm i is an H-independent set of G i, i = 1, 2. Hence β H (G) M = M 1 + M 2 2 = β H (G 1 ) + β H (G 2 ) 2. (i) It immediately follows by the above claims. (ii) Sufficiency: The result follows by Claim 2 and (i). Necessity: By Claim 1, x is no β H -fixed vertex of G. Then there is a β H -set M of G with x M. Assume to the contrary, x is a β H -fixed vertex of G i for some i {1, 2}, say i = 1. Since H is induced-hereditary, M 1 V (G 1 ) is an H-independent set of G 1 x and M 2 = M V (G 2 ) is an H-independent set of G 2. Hence β H (G) = M = M 1 + M 2 β H (G 1 x) + β H (G 2 ) = (β H (G 1 ) 1) + β H (G 2 ) (because of Theorem 2.1 (2)) - a contradiction. (iii) Sufficiency: Let M be a β H -set of G, x M and M i = V (G) M, i = 1, 2. Since H is induced-hereditary, M 1 and M 2 are H-independent sets and since x M, M i is no β H -set of G i, i = 1, 2. This implies β H (G) = M = M 1 + M 2 (β H (G 1 ) 1) + (β H (G 2 ) 1) and the result follows by (i). Necessity: By Claims 2 and 3, it follows that x is a β H -fixed vertex of G i, i = 1, 2. Let M i be a β H -set of G i, i = 1, 2. Since H is additive and inducedhereditary, M = (M 1 M 2 ) {x} is an H-independent set of G with M = M 1 + M 2 2 = β H (G 1 ) + β H (G 2 ) 2 = β H (G). Hence M is an β H -set of G and x M. (iv) Assume to the contrary, β H (G 1 )+β H (G 2 ) 2 β H (G). By (i), β H (G 1 )+ β H (G 2 ) 2 = β H (G) and by (iii), x i is a β H -fixed vertex of G i, i = 1, 2. Let M i be a β H -set of G i, i = 1, 2. Since H is 1-additive, M 1 M 2 is an H-independent set of G. Hence β H (G) M 1 M 2 = M 1 + M 2 1 = β H (G 1 ) + β H (G 2 ) 1 - a contradiction. 3 Lower P-independence number Theorem 3.1. Let G be a nontrivial graph and let H G be nondegenerate and induced-hereditary. Then: (1) V (G, i H ) = {x V (G) : i H (G x) = i H (G) 1}; (2) V + (G, i H ) Fi(G, i H ); (3) B(G, i H ) V 0 (G, i H ).

Proof. Let v V (G), M 2 be an i H -set of G and v M 2. Then M 2 M H S(G v) implying i H (G) i H (G v). Now let M 3 be an i H -set of G v. Then either M 3 or M 3 {v} is a maximal H-independent set of G. Hence i H (G v) + 1 i H (G) and if the equality holds then v is i H -good. Theorem 3.2. Let G = G 1 x G2 and H G be 1-additive and induced-hereditary. (1) Then i H (G) i H (G 1 ) + i H (G 2 ) 1. (2) Let x be an i H -good vertex of G, i H (G) = i H (G 1 ) + i H (G 2 ) 1, let M be an i H -set of G and x M. Then M V (G j ) is an i H -set of G j, j = 1, 2. (3) Let x be an i H -bad vertex of the graph G, i H (G) = i H (G 1 ) + i H (G 2 ) 1 and let M be an i H -set of G. Then there are l, m such that {l, m} = {1, 2}, M V (G l ) is an i H -set of G l, M V (G m ) is an i H -set of G m x, i H (G m x) = i H (G m ) 1 and (M V (G m )) {x} is an i H -set of G m. (4) Let x be an i H -good vertex of graphs G 1 and G 2. Then i H (G) = i H (G 1 ) + i H (G 2 ) 1. If M j is an i H -set of G j, j = 1, 2 and {x} = M 1 M 2 then M 1 M 2 is an i H -set of the graph G; (5) Let x be an i H -bad vertex of G 1 and G 2. Then i H (G) = i H (G 1 ) + i H (G 2 ). If M j is an i H -set of G j, j = 1, 2 then M 1 M 2 is an i H -set of G. Proof. Since H is 1-additive and induced-hereditary, H is nondegenerate. need the following claims: Claim 5. Let M i M H S(G i ), i = 1, 2 and either x M 1 M 2 or {x} = M 1 M 2. Then M 1 M 2 M H S(G). Proof. Since H is 1-additive, M = M 1 M 2 is an H-independent set of G. If M M H S(G) then by Lemma 1.1, there is u V (G M) such that M {u} is an H-independent set of G. Let without loss of generality u V (G 1 ). Since H is induced-hereditary then M 1 {u} is an H-independent set of G 1 contradicting M 1 M H S(G 1 ). Hence M M H S(G). Claim 6. Let x M M H S(G) and M j = M V (G j ), j = 1, 2. Then M j M H S(G j ) for j = 1, 2. Proof. Since H is induced-hereditary, M i is an H-independent set of G i, i = 1, 2. Assume M i M H S(G i ) for some i {1, 2}. By Lemma 1.1, there is u V (G i M i ) such that M i {u} is an H-independent set in G i. Since H is 1- additive, M {u} is an H-independent set of G which contradicts the maximality of M. We

Claim 7. Let x M M H S(G) and M j = M V (G j ), j = 1, 2. Then one of the following holds: 1. M j M H S(G j ) for j = 1, 2; 2. there are l and m such that {l, m} = {1, 2}, M l M H S(G l ), M m M H S(G m x) and M m {x} M H S(G m ). Proof. Since H is induced-hereditary, M i is an H-independent set of G i, i = 1, 2. Suppose there is j {1, 2} such that M j M H S(G j ), say j = 1. If M 1 M H S(G 1 x) then by Lemma 1.1 there is v V (G 1 x), v M 1 such that M 1 {v} is an H-independent set of G 1 x and since H is additive, M {v} is an H-independent set of G - a contradiction. So, M 1 M H S(G 1 x). Since M 1 M H S(G 1 ), by Lemma 1.1 there is u V (G 1 M 1 ) such that M 1 {u} is an H-independent set of G 1. Since M 1 M H S(G 1 x) it follows that u = x. Hence M 1 {x} M H S(G 1 ). Suppose M 2 M H S(G 2 ). Then M 2 {x} M H S(G 2 ) and by Claim 5, M {x} M H S(G) contradicting M M H S(G). (1), (2) and (3): Let M be an i H -set of G. Since H is induced-hereditary then M i = M V (G i ) is an H-independent set of G i, i = 1, 2. Case x M: By Claim 6, M i M H S(G i ), i = 1, 2. Hence i H (G) = M = M 1 + M 2 1 i H (G 1 ) + i H (G 2 ) 1. Clearly the equality holds if and only if M i is an i H -set of G i, i = 1.2. Case x M: If M j M H S(G j ), j = 1, 2 then i H (G) = M = M 1 + M 2 i H (G 1 )+i H (G 2 ). Let without loss of generality, M 1 M H S(G 1 ). From Claim 7 it follows that M 1 M H S(G 1 x), M 1 {x} M H S(G 1 ) and M 2 M H S(G 2 ). Now we have i H (G) = M = M 1 + M 2 i H (G 1 x)+i H (G 2 ) i H (G 1 )+i H (G 2 ) 1 (by Theorem 3.1(1)) and the equality holds if and only if M 2 is an i H -set of G 2, M 1 is an i H -set of G 1 x, M 1 {x} is an i H -set of G 1 and i H (G 1 x) = i H (G 1 ) 1. (4): Let M j be an i H -set of G j, j = 1, 2 and {x} = M 1 M 2. By Claim 5 we have M 1 M 2 M H S(G). Hence i H (G) M 1 M 2 = M 1 + M 2 1 = i H (H 1 )+i H (G 2 ) 1. Now by (1), i H (G) = i H (G 1 )+i H (G 2 ) 1 and then M 1 M 2 is an i H -set of G. (5): Assume i H (G) = i H (G 1 ) + i H (G 2 ) 1. If x is an i H -bad vertex of G then by (3) there exists m {1, 2} such that i H (G m x) = i H (G m ) 1. Now by Theorem 3.1 x is an i H -good vertex of G m - a contradiction. If x is an i H - good vertex of G, M is an i H -set of G and x M then by (2) we have M s is an i H -set of G s, s = 1, 2. But then x is an i H -good vertex of G s, s = 1, 2 which is a contradiction. Hence i H (G) i H (G 1 ) + i H (G 2 ). Let M j be an i H - set of G j, j = 1, 2. Now by Claim 5 it follows M 1 M 2 M H S(G). Hence

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