Minimum r-neighborhood covering set of permutation graphs

Abstract

For a connected graph [Formula: see text] and a fixed integer [Formula: see text], a node [Formula: see text] [Formula: see text]-dominates another node [Formula: see text] if [Formula: see text]. An edge [Formula: see text] is [Formula: see text]-neighborhood covered by a vertex [Formula: see text], if [Formula: see text] and [Formula: see text], i.e., both the vertices [Formula: see text] and [Formula: see text] are [Formula: see text]-dominated by the vertex [Formula: see text]. A set [Formula: see text] is known to be a [Formula: see text]-neighborhood covering ([Formula: see text]-NC) set of graph [Formula: see text] if and only if one or more vertices of [Formula: see text] [Formula: see text]-dominate each edge in E. Among all [Formula: see text]-NC sets of graph [Formula: see text], the set with fewest cardinality is the minimum [Formula: see text]-NC set of [Formula: see text] and we indicate its cardinality as [Formula: see text]-NC-number and we denote it by the symbol [Formula: see text]. This is an NP-complete problem on general graphs. It is also NP-complete for chordal graphs. Here, we develop an [Formula: see text] time algorithm for computing a minimum [Formula: see text]-NC set of permutation graphs, where [Formula: see text] indicates the order of the set [Formula: see text].