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ALMOST-PERIPHERAL GRAPHS

Sandi Klavžar, Kishori P. Narayankar, H. B. Walikar and S. B. Lokesh
Taiwanese Journal of Mathematics
Vol. 18, No. 2 (April 2014), pp. 463-471
Stable URL: http://www.jstor.org/stable/taiwjmath.18.2.463
Page Count: 9
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ALMOST-PERIPHERAL GRAPHS
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Abstract

Abstract The center C(G) and the periphery P(G) of a connected graph G consist of the vertices of minimum and maximum eccentricity, respectively. Almost-peripheral (AP) graphs are introduced as graphs G with |P(G)| = |V(G)| − 1 (and |C(G)| = 1). AP graph of radius r is called an r-AP graph. Several constructions of AP graph are given, in particular implying that for any r ≥ 1, any graph can be embedded as an induced subgraph into some r-AP graph. A decomposition of AP-graphs that contain cut-vertices is presented. The r-embedding index Φr(G) of a graph G is introduced as the minimum number of vertices which have to be added to G such that the obtained graph is an r-AP graph. It is proved that Φ2(G) ≤ 5 holds for any non-trivial graphs and that equality holds if and only if G is a complete graph. 2010 Mathematics Subject Classification: 05C12, 05C75, 90B80. Key words and phrases: Radius, Diameter, Almost-peripheral graph, Self-centered graph.

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