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- W2146264549 abstract "Let G=(V,E) be a finite, simple and connected graph. Let S⊆V, its closed interval I[S] is the set of all vertices lying on a shortest path between any pair of vertices of S. The set S is geodetic if I[S]=V. The eccentricity of a vertex v is the number of edges in the greatest shortest path between v and any vertex w of G. The contour Ct(G) of G is the set formed by vertices v such that no neighbor of v has an eccentricity greater than v. We consider the problem of determining whether the contour of a graph class is geodetic. The diameter diam(G) of G is the maximum eccentricity of the vertices in V. In this work we establish a relation between the diameter and the geodeticity of the contour of a graph. We prove that the contour is geodetic for graphs with diameter k≤4. Furthermore, for every k>4, there is a graph with diameter k and whose contour is not geodetic. We show that the contour is geodetic for bipartite graphs with diameter k≤7, and for any k>7 there is a bipartite graph with diameter k and non-geodetic contour. By applying these results, we solve the open problems mentioned by Cáceres et al. (2008, 2005) [4], [5] namely to decide whether the contour of cochordal graph, parity graph and bipartite graphs are geodetic." @default.
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- W2146264549 date "2013-07-01" @default.
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- W2146264549 title "On the contour of graphs" @default.
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- W2146264549 doi "https://doi.org/10.1016/j.dam.2012.12.024" @default.
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