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An Ore-type condition for arbitrarily vertex decomposable graphs
Electronic Notes in Discrete Mathematics 22 (2005) 477–480 www.elsevier.com/locate/endm
An Ore-type condition for arbitrarily vertex decomposable graphs Antoni Marczyk 1 Faculty of Applied Mathematics AGH University of Science and Technology Al. Mickiewicza 30, 30-059 Krak´ ow, Poland
Abstract A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a1 , ..., ak ) of positive integers such that a1 + ... + ak = n there exists a partition (V1 , ..., Vk ) of the vertex set of G such that for each i ∈ {1, ..., k}, Vi induces a connected subgraph of G on ai vertices. In this paper we show that if G is a 2connected graph of order n with the independence number at most n/2 and such that the degree sum of any pair of nonadjacent vertices is at least n − 3, then G is arbitrarily vertex decomposable. We present a similar result for connected graphs satisfying a similar condition where n − 3 is replaced by n − 2. Keywords: Arbitrarily vertex decomposable graphs, Traceable graphs
1
Introduction
Let G be a simple, undirected graph of order n and let k, 1 ≤ k ≤ n, be an integer. G is said to be k-vertex decomposable if for each sequence (a1 , ..., ak ) of positive integers such that a1 +...+ak = n there exists a partition (V1 , ..., Vk ) 1
Email: [email protected]
1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2005.06.097
478
A. Marczyk / Electronic Notes in Discrete Mathematics 22 (2005) 477–480
of the vertex set of G such that for each i ∈ {1, ..., k}, Vi induces a connected subgraph of G on ai vertices. G is called arbitrarily vertex decomposable (avd for short) if it is k-vertex decomposable for each k ∈ {1, . . . , n}. The problem of describing avd trees has been treated in several papers. Notice that the investigation of trees is motivated by the fact that a connected graph is avd if its spanning tree is avd. In [6] Horˇ n´ak and Wo´zniak conjectured that if T is a tree with maximum degree Δ(T ) at least five, then T is not avd. This conjecture was proved by Barth and Fournier [2]. The first result characterizing nontrivially avd trees (i.e. caterpillars with three leaves) was found by Barth et al. [1] and, independently, by Horˇ n´ak and Wo´zniak [5]. In [1] and [2] Barth et al. and Barth and Fournier studied a family of trees each of them being homeomorphic to K1,3 or K1,4 (they call them tripods or 4-pods) and showed that determining if such a tree is avd can be done using a polynomial algorithm. In [3] Cichacz et al. gave a complete characterization of arbitrarily vertex decomposable caterpillars with four leaves. They also described two families of arbitrarily vertex decomposable trees with maximum degree three or four. There are also some results on avd graphs which can have cycles. Gy˝ori [4] and, independently, Lov´asz [8] proved that every k-connected graph is k-vertex decomposable. In [7] Kalinowski et al. investigated unicyclic avd graphs where the unique cycle is dominating. Generally, the problem of deciding whether a given graph is arbitrarily vertex decomposable is NP-complete [10] but we do not know if this problem is NP-complete when restricted to trees. However, it is obvious that each path and each traceable graph is avd. Therefore, each condition implying the existence of a hamiltonian path in a graph also implies that the graph is avd. So we can try to replace some known conditions for traceability by the weaker ones implying that the graphs satisfying these conditions are avd. Let σ2 (G) := min{d(x) + d(y)| x, y are nonadjacent vertices in G} if G is not a complete graph, and σ2 (G) = ∞ otherwise. It follows by a well-known result of Ore [9] that if G is a graph of order n with σ2 (G) ≥ n − 1 then G is traceable. In this note we will investigate the avd graphs satisfying the condition σ2 (G) ≥ n − 2 or σ2 (G) ≥ n − 3. Observe that any necessary condition for a graph to contain a perfect matching (or a matching that omits exactly one vertex) is a necessary condition for a graph to be arbitrarily vertex decomposable. Thus, we will assume that the independence number α(G) is at most n/2.
A. Marczyk / Electronic Notes in Discrete Mathematics 22 (2005) 477–480
2
479
Main results
In the sequel we denote by G ∨ H the join of two disjoint graphs G and H (obtained by joining all vertices of G to all vertices of H) and by Gp any graph of order p. Theorem 2.1 Let G be a connected graph of order n such that σ2 (G) ≥ n − 2 and α(G) is at most n/2. Then G is avd. Corollary 2.2 If G is a graph of order n with σ2 (G) ≥ n − 2, then G is avd or an union of two disjoint cliques or n is even and G is the join K (n+2)/2 ∨ G(n−2)/2 . Theorem 2.3 Let G be a 2-connected graph such that α(G) ≤ n/2 and σ2 (G) ≥ n − 3. Then G is avd. Corollary 2.4 If G is a 2-connected graph of order n with σ2 (G) ≥ n − 3, then G is avd or n ≥ 7 is odd and G = K (n+3)/2 ∨ G(n−3)/2 or n ≥ 6 is even, G = K (n+2)/2 ∨ G(n−2)/2 , or G = K (n+2)/2 ∨ G(n−2)/2 \ e, where e is an edge which joins V (K (n+2)/2 ) with V (G(n−2)/2 ). Corollary 2.5 If G is a 2-connected graph such that σ2 (G) ≥ n − 3, then for every integer k ∈ / {(n − 1)/2, n/2, (n + 1)/2} G is k-vertex decomposable.
References [1] Barth, D., O. Baudon and J. Puech, Network sharing: a polynomial algorithm for tripods, Discrete Applied Mathematics, 119 (2002) 205-216. [2] Barth, D., and H. Fournier, A Degree Bound on Decomposable Trees, preprint (2004). [3] Cichacz S., A. G¨ orlich, A. Marczyk, J. Przybylo, M. Wo´zniak, Arbitrarily vertex decomposable caterpillars with four or five leaves, preprint 2005. [4] Gy˝ ori, E., ”On division of graphs to connected subgraphs”, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I, pp. 485-494, Colloq. Math. Soc. J´ anos Bolyai, 18, North-Holland, Amsterdam-New York, 1978. [5] Horˇ na´k M., and M. Wo´zniak, On arbitrarily vertex decomposable trees, Technical report, Faculty of Applied Mathematics, Krakow (2003), submitted. [6] Horˇ na´k, M., and M. Wo´zniak, Arbitrarily vertex decomposable trees are of maximum degree at most six, Opuscula Mathematica 23 (2003), 49-62.
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A. Marczyk / Electronic Notes in Discrete Mathematics 22 (2005) 477–480
[7] Kalinowski, R., M. Pil´sniak, M. Wo´zniak, I. Ziolo, Arbitrarily vertex decomposable suns with few rays, preprint 2005. [8] Lov´ asz, L., A homology theory for spanning trees of a graph, Acta Math. Acad. Sci. Hungar. 30 (1977), no. 3-4, 241-251. [9] Ore, O., Note on hamilton circuits, Amer. Math. Monthly 67 (1960), 55. [10] Robson, M., Private communication, Universit´e Bordeaux I (1998).
An Ore-type condition for arbitrarily vertex decomposable graphs Antoni Marczyk 1 Faculty of Applied Mathematics AGH University of Science and Technology Al. Mickiewicza 30, 30-059 Krak´ ow, Poland
Abstract A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a1 , ..., ak ) of positive integers such that a1 + ... + ak = n there exists a partition (V1 , ..., Vk ) of the vertex set of G such that for each i ∈ {1, ..., k}, Vi induces a connected subgraph of G on ai vertices. In this paper we show that if G is a 2connected graph of order n with the independence number at most n/2 and such that the degree sum of any pair of nonadjacent vertices is at least n − 3, then G is arbitrarily vertex decomposable. We present a similar result for connected graphs satisfying a similar condition where n − 3 is replaced by n − 2. Keywords: Arbitrarily vertex decomposable graphs, Traceable graphs
1
Introduction
Let G be a simple, undirected graph of order n and let k, 1 ≤ k ≤ n, be an integer. G is said to be k-vertex decomposable if for each sequence (a1 , ..., ak ) of positive integers such that a1 +...+ak = n there exists a partition (V1 , ..., Vk ) 1
Email: [email protected]
1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2005.06.097
478
A. Marczyk / Electronic Notes in Discrete Mathematics 22 (2005) 477–480
of the vertex set of G such that for each i ∈ {1, ..., k}, Vi induces a connected subgraph of G on ai vertices. G is called arbitrarily vertex decomposable (avd for short) if it is k-vertex decomposable for each k ∈ {1, . . . , n}. The problem of describing avd trees has been treated in several papers. Notice that the investigation of trees is motivated by the fact that a connected graph is avd if its spanning tree is avd. In [6] Horˇ n´ak and Wo´zniak conjectured that if T is a tree with maximum degree Δ(T ) at least five, then T is not avd. This conjecture was proved by Barth and Fournier [2]. The first result characterizing nontrivially avd trees (i.e. caterpillars with three leaves) was found by Barth et al. [1] and, independently, by Horˇ n´ak and Wo´zniak [5]. In [1] and [2] Barth et al. and Barth and Fournier studied a family of trees each of them being homeomorphic to K1,3 or K1,4 (they call them tripods or 4-pods) and showed that determining if such a tree is avd can be done using a polynomial algorithm. In [3] Cichacz et al. gave a complete characterization of arbitrarily vertex decomposable caterpillars with four leaves. They also described two families of arbitrarily vertex decomposable trees with maximum degree three or four. There are also some results on avd graphs which can have cycles. Gy˝ori [4] and, independently, Lov´asz [8] proved that every k-connected graph is k-vertex decomposable. In [7] Kalinowski et al. investigated unicyclic avd graphs where the unique cycle is dominating. Generally, the problem of deciding whether a given graph is arbitrarily vertex decomposable is NP-complete [10] but we do not know if this problem is NP-complete when restricted to trees. However, it is obvious that each path and each traceable graph is avd. Therefore, each condition implying the existence of a hamiltonian path in a graph also implies that the graph is avd. So we can try to replace some known conditions for traceability by the weaker ones implying that the graphs satisfying these conditions are avd. Let σ2 (G) := min{d(x) + d(y)| x, y are nonadjacent vertices in G} if G is not a complete graph, and σ2 (G) = ∞ otherwise. It follows by a well-known result of Ore [9] that if G is a graph of order n with σ2 (G) ≥ n − 1 then G is traceable. In this note we will investigate the avd graphs satisfying the condition σ2 (G) ≥ n − 2 or σ2 (G) ≥ n − 3. Observe that any necessary condition for a graph to contain a perfect matching (or a matching that omits exactly one vertex) is a necessary condition for a graph to be arbitrarily vertex decomposable. Thus, we will assume that the independence number α(G) is at most n/2.
A. Marczyk / Electronic Notes in Discrete Mathematics 22 (2005) 477–480
2
479
Main results
In the sequel we denote by G ∨ H the join of two disjoint graphs G and H (obtained by joining all vertices of G to all vertices of H) and by Gp any graph of order p. Theorem 2.1 Let G be a connected graph of order n such that σ2 (G) ≥ n − 2 and α(G) is at most n/2. Then G is avd. Corollary 2.2 If G is a graph of order n with σ2 (G) ≥ n − 2, then G is avd or an union of two disjoint cliques or n is even and G is the join K (n+2)/2 ∨ G(n−2)/2 . Theorem 2.3 Let G be a 2-connected graph such that α(G) ≤ n/2 and σ2 (G) ≥ n − 3. Then G is avd. Corollary 2.4 If G is a 2-connected graph of order n with σ2 (G) ≥ n − 3, then G is avd or n ≥ 7 is odd and G = K (n+3)/2 ∨ G(n−3)/2 or n ≥ 6 is even, G = K (n+2)/2 ∨ G(n−2)/2 , or G = K (n+2)/2 ∨ G(n−2)/2 \ e, where e is an edge which joins V (K (n+2)/2 ) with V (G(n−2)/2 ). Corollary 2.5 If G is a 2-connected graph such that σ2 (G) ≥ n − 3, then for every integer k ∈ / {(n − 1)/2, n/2, (n + 1)/2} G is k-vertex decomposable.
References [1] Barth, D., O. Baudon and J. Puech, Network sharing: a polynomial algorithm for tripods, Discrete Applied Mathematics, 119 (2002) 205-216. [2] Barth, D., and H. Fournier, A Degree Bound on Decomposable Trees, preprint (2004). [3] Cichacz S., A. G¨ orlich, A. Marczyk, J. Przybylo, M. Wo´zniak, Arbitrarily vertex decomposable caterpillars with four or five leaves, preprint 2005. [4] Gy˝ ori, E., ”On division of graphs to connected subgraphs”, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I, pp. 485-494, Colloq. Math. Soc. J´ anos Bolyai, 18, North-Holland, Amsterdam-New York, 1978. [5] Horˇ na´k M., and M. Wo´zniak, On arbitrarily vertex decomposable trees, Technical report, Faculty of Applied Mathematics, Krakow (2003), submitted. [6] Horˇ na´k, M., and M. Wo´zniak, Arbitrarily vertex decomposable trees are of maximum degree at most six, Opuscula Mathematica 23 (2003), 49-62.
480
A. Marczyk / Electronic Notes in Discrete Mathematics 22 (2005) 477–480
[7] Kalinowski, R., M. Pil´sniak, M. Wo´zniak, I. Ziolo, Arbitrarily vertex decomposable suns with few rays, preprint 2005. [8] Lov´ asz, L., A homology theory for spanning trees of a graph, Acta Math. Acad. Sci. Hungar. 30 (1977), no. 3-4, 241-251. [9] Ore, O., Note on hamilton circuits, Amer. Math. Monthly 67 (1960), 55. [10] Robson, M., Private communication, Universit´e Bordeaux I (1998).