4 research outputs found
Krausz dimension and its generalizations in special graph classes
A {\it krausz -partition} of a graph is the partition of into
cliques, such that any vertex belongs to at most cliques and any two
cliques have at most vertices in common. The {\it -krausz} dimension
of the graph is the minimum number such that has a
krausz -partition. 1-krausz dimension is known and studied krausz
dimension of graph .
In this paper we prove, that the problem is polynomially
solvable for chordal graphs, thus partially solving the problem of P. Hlineny
and J. Kratochvil. We show, that the problem of finding -krausz dimension is
NP-hard for every , even if restricted to (1,2)-colorable graphs, but
the problem is polynomially solvable for -polar
graphs for every fixed
A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs
We characterize the class L32 of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from L32 in the class of threshold graphs, where n is the number of vertices of a tested graph
Krausz dimension and its generalizations in special graph classes
Graph TheoryA 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
A finite characterization and recognition of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 in the class of thresholds graphs
International audience2010 Mathematics Subject Classification: 05C62, 05C75, 05C70, 05C65, 05C8