Linear time determination of the scattering number for strictly chordal graphs

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.