Uniformly‐most reliable networks do not always exist

Myrvold, Wendy; Cheung, Kim H.; Page, Lavon B.; Perry, Jo Ellen

doi:10.1002/net.3230210404

Cited by 75 publications

(52 citation statements)

References 2 publications

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“…Thus, given such an orientation, an algorithm implementing our contraction approach may lead to efficient solutions for other combinatorial enumeration problems. Further, as mentioned in the introduction, a uniformly-most reliable network (defined in [11,27]) must maximize the number of spanning trees. Thus, it is interesting to determine the types of graphs which have the maximum number of spanning trees for fixed numbers of vertices and edges (see [29,33]).…”

Section: Discussionmentioning

confidence: 99%

Counting spanning trees using modular decomposition

Nikolopoulos

Palios

Papadopoulos

2014

Theoretical Computer Science

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In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process. In particular, when applied on a (q, q − 4)-graph for fixed q, a P 4 -tidy graph, or a tree-cograph, our algorithm computes the number of its spanning trees in time linear in the size of the graph, where the complexity of arithmetic operations is measured under the uniform-cost criterion. Therefore we give the first linear-time algorithm for the counting problem in the considered graph classes.

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Section: Discussionmentioning

confidence: 99%

Counting spanning trees using modular decomposition

Nikolopoulos

Palios

Papadopoulos

2014

Theoretical Computer Science

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show abstract

“…Thus, given such an orientation, an algorithm implementing our contraction approach may lead to efficient solutions for other combinatorial enumeration problems. Further, as mentioned in the introduction, a uniformly-most reliable network (defined in [8,19]) must maximize the number of spanning trees. Thus, it is interesting to determine the types of graphs which have the maximum number of spanning trees for fixed numbers of vertices and edges (see [21,25]).…”

Section: Discussionmentioning

confidence: 99%

Counting Spanning Trees in Graphs Using Modular Decomposition

Nikolopoulos

Palios

Papadopoulos

2011

WALCOM: Algorithms and Computation

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Abstract. In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process. In particular, when applied on a (q, q − 4)-graph for fixed q, a P4-tidy graph, or a tree-cograph, our algorithm computes the number of its spanning trees in time linear in the size of the graph, where the complexity of arithmetic operations is measured under the uniform-cost criterion. Therefore we give the first linear-time algorithm for the counting problem in the considered graph classes.

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“…, r), such as k 1 and k 3 , which may be a member l-min m l/1 graphs without containing any nearly indeof some 2-chain-cutset of G 0 k t . It is easy to see that pendent four-edge disconnecting set, excepting (n, e) Å (7, 11) and (8,12).…”

Section: Max L-min M L/1 Graphs For 2e/n ¢mentioning

confidence: 98%