Krausz dimension and its generalizations in special graph classes

TL;DR

This paper studies the Krausz dimension and its generalizations in special graph classes, proving polynomial solvability for certain cases and NP-hardness for others, advancing understanding of graph partition problems.

Contribution

It proves polynomial algorithms for Krausz dimension problems in chordal and $(\infty,1)$-polar graphs, and NP-hardness results for general cases, extending prior work.

Findings

Polynomial solvability of $kdim(G) \,\leq 3$ for chordal graphs.

NP-hardness of finding $m$-Krausz dimension for $m\geq 1$ in general.

Polynomial solvability of $kdim_m(G)\leq k$ in $(\infty,1)$-polar graphs.

Abstract

A {\it krausz $(k,m)$-partition} of a graph $G$ is the partition of $G$ into cliques, such that any vertex belongs to at most $k$ cliques and any two cliques have at most $m$ vertices in common. The {\it $m$-krausz} dimension $kdim_m(G)$ of the graph $G$ is the minimum number $k$ such that $G$ has a krausz $(k,m)$-partition. 1-krausz dimension is known and studied krausz dimension of graph $kdim(G)$. In this paper we prove, that the problem $"kdim(G)\leq 3"$ is polynomially solvable for chordal graphs, thus partially solving the problem of P. Hlineny and J. Kratochvil. We show, that the problem of finding $m$-krausz dimension is NP-hard for every $m\geq 1$, even if restricted to (1,2)-colorable graphs, but the problem $"kdim_m(G)\leq k"$ is polynomially solvable for $(\infty,1)$-polar graphs for every fixed $k,m\geq 1$.