Abstract
Graphs and Algorithms A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.
Highlights
All graphs we consider are finite, simple, and undirected, and for a graph G = (V, E) we use n = |V | and m = |E| to refer to the number of vertices and edges of the graph
We study classes of graphs that have a polynomial bound on the number of maximal cliques
Given a Helly intersection representation of polynomial size for a class of graphs, the maximum weight clique problem can be solved in polynomial time for the intersection class [18] as well as for the corresponding overlap graph class [4]
Summary
A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms.
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