Linear time determination of the scattering number for strictly chordal graphs

TL;DR

This paper presents a linear time algorithm for computing the scattering number and scattering set in strictly chordal graphs, a subclass of chordal graphs, enhancing understanding of their vulnerability measures.

Contribution

It introduces the first linear time solution for determining the scattering number specifically for strictly chordal graphs.

Findings

Linear time algorithm for scattering number in strictly chordal graphs

Scattering number computation is not solely determined by toughness

Enhanced understanding of graph vulnerability measures

Abstract

The scattering number of a graph $G$ was defined by Jung in 1978 as $sc(G) = max \{\omega(G - S) - |S|, S \subseteq V, \omega(G - S) \neq1\}$ where $\omega(G - S) $ is the number of connected components of the graph $G-S$. It is a measure of vulnerability of a graph and it has a direct relationship with the toughness of a graph. Strictly chordal graphs, also known as block duplicate graphs, are a subclass of chordal graphs that includes block and 3-leaf power graphs. In this paper we present a linear time solution for the determination of the scattering number and scattering set of strictly chordal graphs. We show that, although the knowledge of the toughness of the class is helpful, it is not sufficient to provide an immediate result for determining the scattering number.