On Zero Forcing Number of Permutation Graphs

Abstract

Zero forcing number, Z(G), of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in $V(G)\!\setminus\!S$ are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the “AIM Minimum Rank – Special Graphs Work Group”. Let G ₁ and G ₂ be disjoint copies of a graph G and let σ: V(G ₁) → V(G ₂) be a permutation. Then a permutation graph G _σ  = (V, E) has the vertex set V = V(G ₁) ∪ V(G ₂) and the edge set E = E(G ₁) ∪ E(G ₂) ∪ {uv |v = σ(u)}. It is readily seen that 1 + δ(G) ≤ Z(G _σ ) ≤ n, if G is a graph of order n ≥ 2; here δ(G) is the minimum degree of G. We give examples showing that |Z(G) − Z(G _σ )| can be arbitrarily large. Further, we characterize permutation graphs G _σ satisfying Z(G _σ ) = n for a graph G that is a nearly complete graph, a complete k-partite graph, a cycle, and a path, respectively, on n vertices.