P3-convexity on graphs with diameter two: Computing hull and interval numbers

Abstract

Given a finite, simple, and undirected graph G = ( V , E ) , a P 3 -convex set of G is a set S ⊆ V ( G ) such that every vertex of V ( G ) ∖ S has at most one neighbor in S , and the P 3 -convex hull of a vertex subset S ′ of G is the smallest P 3 -convex set containing S ′ . The P 3 -hull number h ( G ) is the cardinality of the smallest set of vertices whose P 3 -convex hull is V ( G ) . In this paper, we establish some bounds on the P 3 -hull number of graphs with diameter two. Particularly, for a biconnected diameter-two graph G it holds that h ( G ) ≤ ⌈ log ( Δ ( G ) + 1 ) ⌉ + 1 , while a non-biconnected diameter-two graph G has h ( G ) = c c ( G − v ) , where v is a cut vertex of G and c c ( G − v ) is the number of connected components of G [ V ∖ { v } ] . In addition, we show that the P 3 -hull number of biconnected C 6 -free diameter-two graphs is at most 4, while for strongly regular graphs G = G ( n , k , b , c ) it holds that h ( G ) ≤ min { ⌈ k 1 + b ⌉ + 1 , ⌈ log c + 1 ( k . c + 1 ) ⌉ + 1 } . Finally, we consider the P 3 -interval number, which is the cardinality of the smallest set S of vertices such that every vertex not in S has two neighbors in this set. We show that computing the P 3 -interval number on diameter-two split graphs is NP - complete .