Hull number: P5-free graphs and reduction rules

Description

In this paper, we study the (geodesic) hull number of graphs. For any two vertices $u,v\in V$ of a connected undirected graph $G=(V,E)$, the closed interval $I[u,v]$ of $u$ and $v$ is the set of vertices that belong to some shortest $(u,v)$-path. For any $S \subseteq V$, let $I[S]= \bigcup_{u,v\in S} I[u,v]$. A subset $S\subseteq V$ is (geodesically) convex if $I[S] = S$. Given a subset $S\subseteq V$, the convex hull $I_h[S]$ of $S$ is the smallest convex set that contains $S$. We say that $S$ is a hull set of $G$ if $I_h[S] = V$. The size of a minimum hull set of $G$ is the hull number of $G$, denoted by $hn(G)$. First, we show a polynomial-time algorithm to compute the hull number of any $P_5$-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph $G$, where the parameter can be the size of a vertex cover of $G$ or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

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