Subtree filament graphs are the intersection graphs of subtree filaments in a tree. This class of graphs contains subtree overlap graphs, interval filament graphs, chordal graphs, circle graphs, circular-arc graphs, cocomparability graphs, and polygon-circle graphs. In this paper we show that, for circle graphs, the clique cover problem is NP-complete and the h-clique cover problem for fixed h ...

In this paper, we show that any B2-VPG graph (i.e., an intersection graph of orthogonal curves with at most 2 bends) can be decomposed into O(logn) outerstring graphs or O(log n) permutation graphs. This leads to better approximation algorithms for hereditary graph problems, such as independent set, clique and clique cover, on B2-VPG graphs.

In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of m labels is chosen independently with probability p by each ...

Let α(G) and β(G), denote the size of a largest independent set and the clique cover number of an undirected graph G. Let H be an interval graph with V (G) = V (H) and E(G) ⊆ E(H), and let φ(G,H) denote the maximum of β(G[W ]) α(G[W ]) overall induced subgraphs G[W ] of G that are cliques in H . The main result of this paper is to prove that for any graph G β(G) ≤ 2α(H)φ(G,H)(logα(H) + 1), wher...

Several new tools are presented for determining the number of cliques needed to (edge-)partition a graph . For a graph on n vertices, the clique partition number can grow cn z times as fast as the clique covering number, where c is at least 1/64. If in a clique on n vertices, the edges between en° vertices are deleted, Z--a < 1, then the number of cliques needed to partition what is left is asy...

Let G be a graph on n vertices and let G be its complement in K,,, the complete graph on n vertices . If f is a real valued function defined on graphs, what are the extreme values of f (G) +f (G) and f (G) f (G)? E. A. Nordhaus and J . W. Gaddum (see e .g . [5]) considered those questions when the function is the chromatic number . D . Taylor, R. D . Dutton and R. C. Brigham [5] studied the que...

Suppose G=( V,E) is a graph in which each maximal clique Ci is associated with an integer ri, where 0 < ri < ICi 1. The generalized clique transversal problem is to determine the minimum cardinality of a subset D of V such that ID n Cij >ri for every maximal clique Ci of G. The problem includes the clique-transversal problem, the i, 1 clique-cover problem, and for perfect graphs, the maximum q-...

Let a set of n fixed length intervals and a set of n (larger) windows, in one-to-one correspondence with the intervals, be given, and assume that each interval can be placed in any position within its window. If the position of each interval has been fixed, the intersection graph of such set of intervals is an interval graph. By varying the position of each interval in all possible ways, we get...