Abstract
The scattering number and isolated scattering number of a graph have been introduced in relation to Hamiltonian properties and network vulnerability, and the isolated scattering number plays an important role in characterizing graphs with a fractional 1-factor. Here we investigate the computational complexity of one variant, namely, the weighted isolated scattering number. We give a polynomial time algorithm to compute this parameter of interval graphs, an important subclass of perfect graphs.
Highlights
Throughout this paper, we use Bondy and Murty [1] for terminology and notations not defined here, and we consider finite simple undirected graphs only
Given an interval graph G = (V, E), consider a point x on the part of the real line covered by the intervals attached to the vertices of G
The problem of finding the weighted isolated scattering number of an interval graph G reduces to finding a attaining cut X of G such that X can be expressed as a union of local cuts, as generated by Algorithm 1
Summary
Throughout this paper, we use Bondy and Murty [1] for terminology and notations not defined here, and we consider finite simple undirected graphs only. The weighted isolated scattering number of a noncomplete graph G = (V, E) is defined as iscw (G) = max {i (G − X) − w (X) : X ∈ C (G)} , (3). We start with some additional definitions and notation and present some preliminary results that are used to characterize properties of an attaining cut and give a formula for computing the weighted isolated scattering number of a noncomplete connected interval graph from its minimal local cuts. A polynomial algorithm for computing the weighted isolated scattering number of an interval graph was presented, which is based on dynamic programming on segments. This is followed by a correctness proof and an analysis of the time complexity.
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