592 research outputs found

    Matching cutsets in graphs of diameter 2

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    AbstractWe say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue, and every vertex has at most one neighbour coloured in the opposite colour. In this paper we study the algorithmic complexity of a problem of recognizing graphs which possess a matching cutset. In particular we present a polynomial-time algorithm which solves this problem for graphs of diameter two

    Growth and isoperimetric profile of planar graphs

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    Let G be a planar graph such that the volume function of G satisfies V(2n)< CV(n) for some constant C > 0. Then for every vertex v of G and integer n, there is a domain \Omega such that B(v,n) \subset \Omega, \Omega \subset B(v, 6n) and the size of the boundary of \Omega is at most order n.Comment: 8 page

    The leafage of a chordal graph

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    The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n - lg n - (1/2) lg lg n + O(1). The proper leafage l*(G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l*(G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.Comment: 19 pages, 3 figure

    On Cyclic Edge-Connectivity of Fullerenes

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    A graph is said to be cyclic kk-edge-connected, if at least kk edges must be removed to disconnect it into two components, each containing a cycle. Such a set of kk edges is called a cyclic-kk-edge cutset and it is called a trivial cyclic-kk-edge cutset if at least one of the resulting two components induces a single kk-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this article it is shown that a fullerene FF containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that FF has a Hamilton cycle, and as a consequence at least 152n2015\cdot 2^{\lfloor \frac{n}{20}\rfloor} perfect matchings, where nn is the order of FF.Comment: 11 pages, 9 figure
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