The forcing connected detour number of a graph

• Santhakumaran, A. P. ^[1] ; Athisayanathan, S. ^[2]
  1. [1] Hindustan University

     Hindustan University

     India

     
  2. [2] St Xavier’s College

     St Xavier’s College

     India

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 33, Nº. 2, 2014, págs. 147-155
• Idioma: inglés
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• Resumen

  • For two vertices u and v in a graph G = (V, E), the detour distance D(u, v) is the length of a longest u—v path in G.A u—v path of length D(u, v) is called a u—v detour. A set ⊆ V is called a detour set of G if every vertex in G lies on a detour joining a pair of vertices of S.The detour number dn(G) of G is the minimum order of its detour sets and any detour set of order dn(G) is a detour basis of G.A set ⊆ V is called a connected detour set of G if S is detour set of G and the subgraph G[S] induced by S is connected. The connected detour number cdn(G) of G is the minimum order of its connected detour sets and any connected detour set of order cdn(G) is called a connected detour basis of G.A subset T of a connected detour basis S is called a forcing subset for S if S is theuniquecon-nected detour basis containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected detour number of S, denoted by fcdn(S), is the cardinality of a minimum forcing subset for S. The forcing connected detour number of G, denoted by fcdn(G),is fcdn(G) = min{fcdn(S)},where the minimum is taken over all connected detour bases S in G. The forcing connected detour numbers ofcertain standard graphs are obtained. It is shown that for each pair a, b of integers with 0 ≤ a