共查询到20条相似文献,搜索用时 15 毫秒
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Irène Charon 《Discrete Applied Mathematics》2006,154(8):1246-1253
Consider an oriented graph G=(V,A), a subset of vertices C⊆V, and an integer r?1; for any vertex v∈V, let denote the set of all vertices x such that there exists a path from x to v with at most r arcs. If for all vertices v∈V, the sets are all nonempty and different, then we call C an r-identifying code. We describe a linear algorithm which gives a minimum 1-identifying code in any oriented tree. 相似文献
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Changping Wang 《Discrete Applied Mathematics》2007,155(11):1497-1505
Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where EG(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γSS(G). In this paper, a sharp upper bound of γSS(G×H) is presented. 相似文献
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This paper studies a variation of domination in graphs called rainbow domination. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2,…,k} such that for any vertex v with f(v)=0? we have ∪u∈NG(v)f(u)={1,2,…,k}. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G, that is the minimum value of ∑v∈V(G)|f(v)| where f runs over all k-rainbow dominating functions of G. In this paper, we prove that the k-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We then give a linear-time algorithm for the k-rainbow domination problem on trees. For a given tree T, we also determine the smallest k such that . 相似文献
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Let G=(V,E) be a connected graph of order n, t a real number with t?1 and M⊆V(G) with . In this paper, we study the problem of some long paths to maintain their one or two different endpoints in M. We obtain the following two results: (1) for any vertex v∈V(G), there exists a vertex u∈M and a path P with the two endpoints v and u to satisfy , , dG(u)+1-t}; (2) there exists either a cycle C to cover all vertices of M or a path P with two different endpoints u0 and up in M to satisfy , where . 相似文献
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Let G be a graph and S⊆V(G). For each vertex u∈S and for each v∈V(G)−S, we define to be the length of a shortest path in 〈V(G)−(S−{u})〉 if such a path exists, and ∞ otherwise. Let v∈V(G). We define if v⁄∈S, and wS(v)=2 if v∈S. If, for each v∈V(G), we have wS(v)≥1, then S is an exponential dominating set. The smallest cardinality of an exponential dominating set is the exponential domination number, γe(G). In this paper, we prove: (i) that if G is a connected graph of diameter d, then γe(G)≥(d+2)/4, and, (ii) that if G is a connected graph of order n, then . 相似文献
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Francesca Fiorenzi 《Discrete Applied Mathematics》2011,159(17):2045-2049
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An equivalence graph is a disjoint union of cliques, and the equivalence number of a graph G is the minimum number of equivalence subgraphs needed to cover the edges of G. We consider the equivalence number of a line graph, giving improved upper and lower bounds: . This disproves a recent conjecture that is at most three for triangle-free G; indeed it can be arbitrarily large.To bound we bound the closely related invariant σ(G), which is the minimum number of orientations of G such that for any two edges e,f incident to some vertex v, both e and f are oriented out of v in some orientation. When G is triangle-free, . We prove that even when G is triangle-free, it is NP-complete to decide whether or not σ(G)≤3. 相似文献
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Kenji Kimura 《Discrete Applied Mathematics》2010,158(18):2071-2074
We extend the notion of a defensive alliance to weighted graphs. Let (G,w) be a weighted graph, where G is a graph and w be a function from V(G) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑x∈NG(v)∩Sw(x)+w(v)≥∑x∈NG(v)−Sw(x) holds for every vertex v in S. Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most . In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n. Then we prove that for any weight function w on V(G), (G,w) has a defensive weighted alliance of order at most . We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3]. 相似文献
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Zeev Nutov 《Discrete Applied Mathematics》2007,155(2):82-91
Let G be a simple digraph. The dicycle packing number of G, denoted νc(G), is the maximum size of a set of arc-disjoint directed cycles in G. Let G be a digraph with a nonnegative arc-weight function w. A function ψ from the set C of directed cycles in G to R+ is a fractional dicycle packing of G if ∑e∈C∈Cψ(C)?w(e) for each e∈E(G). The fractional dicycle packing number, denoted , is the maximum value of ∑C∈Cψ(C) taken over all fractional dicycle packings ψ. In case w≡1 we denote the latter parameter by .Our main result is that where n=|V(G)|. Our proof is algorithmic and generates a set of arc-disjoint directed cycles whose size is at least νc(G)-o(n2) in randomized polynomial time. Since computing νc(G) is an NP-Hard problem, and since almost all digraphs have νc(G)=Θ(n2) our result is a FPTAS for computing νc(G) for almost all digraphs.The result uses as its main lemma a much more general result. Let F be any fixed family of oriented graphs. For an oriented graph G, let νF(G) denote the maximum number of arc-disjoint copies of elements of F that can be found in G, and let denote the fractional relaxation. Then, . This lemma uses the recently discovered directed regularity lemma as its main tool.It is well known that can be computed in polynomial time by considering the dual problem. We present a polynomial algorithm that finds an optimal fractional dicycle packing. Our algorithm consists of a solution to a simple linear program and some minor modifications, and avoids using the ellipsoid method. In fact, the algorithm shows that a maximum fractional dicycle packing with at most O(n2) dicycles receiving nonzero weight can be found in polynomial time. 相似文献