Partial domination in supercubic graphs

Abstract

For some α with 0<α≤1, a subset X of vertices in a graph G of order n is an α-partial dominating set of G if the set X dominates at least α×n vertices in G. The α-partial domination number pdα(G) of G is the minimum cardinality of an α-partial dominating set of G. In this paper partial domination of graphs with minimum degree at least 3 is studied. It is proved that if G is a graph of order n and with δ(G)≥3, then pd78(G)≤13n. If in addition n≥60, then pd910(G)≤13n, and if G is a connected cubic graph of order n≥28, then pd1314(G)≤13n. Along the way it is shown that there are exactly four connected cubic graphs of order 14 with domination number 5.