Generalized Complements of a Graph

Venkatachalam, C V and Bhat , Pradeep G (1998) Generalized Complements of a Graph. Indian Journal of Pure and Applied Mathematics . pp. 625-639. ISSN 0019-5588

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Abstract

Let G = (V, E) be a graph and P = {VI, V2, ... , VtLbe a partition of V of order k ~ 1. For each set v; in P. remove the edges of G inside Vr and add the edges G, (the complement of G) joining the vertices Yr. The graph Gr (i) thus obtained is called the k(i)-complement of G with respect io P-.The graph G is k(/)-self complementary (k(/)-s.c) if of (I) :: G for .some partition P of V of order k. Further, G is k(1)-co-self complementary (k(/)-co-s.c.) if of (i) :: G. We determine (I) all k(l)-s.c trees for k = 2, 3, and (2) 2(i)-s.c. unicyclic graphs. Also, some necessary conditions for a tree/unicyclic graph to be k(1)-s.c. are obtained. We indicate how to obtain characterizations of all k(i)-co.s.c. trees, unicyclic graphs and forests from known results.

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