# Krausz dimension and its generalizations in special graph classes

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1 Department of Mechanics and Mathematics Minsk

Abstract : A Krausz k,m-partition of a graph G is a decomposition 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 m-Krausz dimension kdimmG of the graph G is the minimum number k such that G has a Krausz k,m-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdimG is a well-known graph-theoretical parameter. In this paper we prove that the problem -kdimG≤3- is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem -kdimmG≤k- is is fixed-parameter tractable when parameterized by k and m for ∞,1-polar graphs. Moreover, the class of ∞,1-polar graphs with kdimmG≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.

Author: ** Olga Glebova - Yury Metelsky - Pavel Skums - **

Source: https://hal.archives-ouvertes.fr/