Irredundancy in Circular Arc Graphs

Golumbic, M.C. and R.C. Laskar, Irredundancy in circular arc graphs, Discrete Applied Mathematics 44 (1993) 79-89. A set ofvertices Xis called irredundant if for every x in Xthe closed neighborhood N[x] contains a vertex which is not a member of N[X-x], the union of the closed neighborhoods of the other vertices. In this paper we show that for circular arc graphs the size of the maximum irredundant set equals the size of a maximum independent set. Variants of irredundancy called oo-irredundance, co-irredundance, and oc-irredundancy are defined using combinations of open and closed neighborhoods. We prove that for circular arc graphs the size of a maximum oo-irredundant set equals 28’ or 2p+l (depending on parity) where/I* is the strong matching number. We also show that for circular arc graphs, the size of a maximum co-irredundant set equals the maximum number of vertices in a set consisting of disjoint K,‘s and Kz’s. Similar results are proven for bipartite graphs.