Identification to Subclasses of Chordal Graphs

TL;DR

This paper investigates the complexity of transforming graphs into subclasses of chordal graphs through vertex identifications, providing a comprehensive analysis for various parameters.

Contribution

It characterizes the classical and parameterized complexity of the Hentity problem for different subclasses of chordal graphs, advancing understanding of graph identification.

Findings

Determined complexity for Hentity with respect to parameters k and n-k.

Analyzed the Hentity problem for various subclasses of chordal graphs.

Established parameterized complexity results for identifying graphs to a target graph H.

Abstract

An identification of two vertices $u$ and $v$ in a graph replaces them with a new vertex whose neighborhood is the union of the neighborhoods of $u$ and $v$. We study the {\sc ${\cal H}$-Identification} problem, which is to decide whether a given graph $G$ can be transformed (``identified'') to a graph in ${\cal H}$ by applying at most $k$ vertex identifications. We determine the classical and parameterized complexity of this problem for various subclasses ${\cal H}$ of chordal graphs, obtaining an almost complete picture for two parameters: $k$ and $n-k$. We also consider the {\sc Identification} problem, which is to test for two given graphs $G$ and $H$ if $G$ can be identified to $H$. We determine the parameterized complexity of this problem when $H$ is a graph from one of our testbed classes, taking the number of simplicial vertices of $H$ as the parameter.