Crossing-Critical Graphs and Path-Width
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منابع مشابه
Structure and generation of crossing-critical graphs
We study c-crossing-critical graphs, which are the minimal graphs that require at least c edgecrossings when drawn in the plane. For c = 1 there are only two such graphs without degree-2 vertices, K5 and K3,3, but for any fixed c > 1 there exist infinitely many c-crossing-critical graphs. It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bou...
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A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number k is that they have bounded bandwidth. A weaker well-known conjecture is that their maximum degree is bounded in terms of k. In this note we disprove these conjectures for every k ≥ 171, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree. A graph is k-crossing-cr...
متن کاملCrossing-number critical graphs have bounded path-width
The crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr(G − e) < cr(G) for all edges e of G. We prove that crossing-critical graphs have “bounded path-width” (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined i...
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The minimum crossing number problem is among the oldest and most fundamental problems arising in the area of automatic graph drawing. In this paper, eight population-based meta-heuristic algorithms are utilized to tackle the minimum crossing number problem for two special types of graphs, namely complete graphs and complete bipartite graphs. A 2-page book drawing representation is employed for ...
متن کاملCrossing Numbers and Cutwidths
The crossing number of a graph G = (V,E), denoted by cr(G), is the smallest number of edge crossings in any drawing of G in the plane. Wee assume that the drawing is good, i.e., incident edges do not cross, two edges cross at most once and at most two edges cross in a point of the plane. Leighton [13] proved that for any n-vertex graph G of bounded degree, its crossing number satisfies cr(G)+n ...
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تاریخ انتشار 2001