Abstract
Graph Theory 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 kdimm(G) 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 kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤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 "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.
Highlights
In this paper we consider finite undirected graphs without loops and multiple edges
The vertex and the edge sets of a graph G are denoted by V (G) and E(G) respectively
Let G(X) denote the subgraph of G induced by a set X ⊆ V (G) and eccG(v) is the eccentricity of a vertex v ∈ V (G)
Summary
In this paper we consider finite undirected graphs without loops and multiple edges. The vertex and the edge sets of a graph (hypergraph) G are denoted by V (G) and E(G) respectively. The class of line graphs has been studied for a long time It is characterized by a finite list of forbidden induced subgraphs [1], efficient algorithms for solving the problem KDIM (2) and constructing the corresponding Krausz 2-partition are known [6, 12, 18, 19]. Denote by KDIMm the problem of determining the m-Krausz dimension of graph, by KDIMm(k) the problem of determining whether kdimm(G) ≤ k and by Lm k the class of graphs with a Krausz (k, m)partition It was proved in [11] that the class Lm 3 could not be characterized by a finite set of forbidden induced subgraphs for every m ≥ 2, but the complexity of the problem KDIMm for an arbitrary m has not been established yet. That the class of (∞, 1)-polar graphs could be recognized in polynomial time [5]
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