Discussiones Mathematicae Graph Theory 32(2) (2012) 299-319
doi: 10.7151/dmgt.1600

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Domination in Functigraphs

Linda Eroh1, Ralucca Gera2, Cong X. Kang 3,
Craig E. Larson4 and Eunjeong Yi3

1 Department of Mathematics
University of Wisconsin Oshkosh
Oshkosh, WI 54901, USA
2 Department of Applied Mathematics
Naval Postgraduate School

Abstract

Let G1 and G2 be disjoint copies of a graph G, and let f:V(G1) → V(G2) be a function. Then a functigraph C(G, f) = (V, E) has the vertex set V = V(G1) ∪V(G2) and the edge set E = E(G1) ∪E(G2) ∪ {uv   |  u ∈ V(G1), v ∈ V(G2),v = f(u)}. A functigraph is a generalization of a permutation graph (also known as a generalized prism) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let γ(G) denote the domination number of G. It is readily seen that γ(G) ≤ γ(C(G,f)) ≤ 2 γ(G). We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.

Keywords: domination, permutation graphs, generalized prisms, functigraphs

2010 Mathematics Subject Classification: 05C69, 05C38.

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Received 12 July 2010
Revised 6 June 2011
Accepted 6 June 2011