Maximum Clique Size Research Articles

On the interval number of a chordal graph

AbstractThe interval number of a (simple, undirected) graph G is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t real intervals. A chordal (or triangulated) graph is one with no induced cycles on 4 or more vertices. If G is chordal and has maximum clique size ω(G) = m, then i(G) ⩽ [1 + o(1)]m/log2 m and this result is best possible, even for split graphs (chordal graphs whose complement is also chordal).

Recognizing planar perfect graphs

An O ( n 3 ) algorithm for recognizing planar graphs that do not contain induced odd cycles of length greater than 3 (odd holes) is presented. A planar graph with this property satisfies the requirement that its maximum clique size equal the minimum number of colors required for the graph (graphs all of whose induced subgraphs satisfy the latter property are perfect as defined by Berge). The algorithm presented is based on decomposing these graphs into essentially two special classes of inseparable component graphs that are easy to recognize. They are (i) planar comparability graphs and (ii) planar line graphs of those planar bipartite graphs whose maximum degrees are no greater than 3. Composition schemes for generating planar perfect graphs from those basic components are also provided. This decomposition algorithm can also be adapted to solve the corresponding maximum independent set and minimum coloring problems. Finally, the path-parity problem on planar perfect graphs is considered.

Maximum induced trees in graphs

Let t( G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investigate the relationship of t( G) to other parameters associated with G: the number of vertices and edges, the radius, the independence number, maximum clique size and connectivity. The central result is a set of upper and lower bounds for the function f( n, ϱ), defined to be the minimum of t( G) over all connected graphs with n vertices and n − 1′ + ϱ edges. The bounds obtained yield an asymptotic characterization of the function correct to leading order in almost all ranges. The results show that f( n, ϱ) is surprisingly small; in particular f( n, cn) = 2 loglog n + O(logloglog n) for any constant c > 0, and f(n, n 1 + γ) = 2 log(1 + 1 γ ) ± 4 for 0 < γ < 1 and n sufficiently large. Bounds on t( G) are obtained in terms of the size of the largest clique. These are used to formulate bounds for a Ramsey-type function, N( k, t), the smallest integer so that every connected graph on N( k, t) vertices has either a clique of size k or an induced tree of size t. Tight bounds for t( G) from the independence number α( G) are also proved. It is shown that every connected graph with radius r has an induced path, and hence an induced tree, on 2 r − 1 vertices.

Ki-covers I: Complexity and polytopes

A K i in a graph is a complete subgraph of size i. A K i -cover of a graph G( V, E is a set C of K i − 1 's of G such that every K i in G contains at least one K i − 1 in C. Thus a K 2-cover is a vertex cover. The problem of determining whether a graph has a K i -cover ( i ⩾ 2) of cardinality ⩽ k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the K i -cover polytope. Further results of this type are also produced for the K 3-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the K i -cover polytope.

Isomorphism Testing in Hookup Classes

Hookup classes are classes of graphs with a certain type of recursive definition, which can be viewed as a generalization of k-trees. We show that many hookup classes of graphs are isomorphism complete, and give polynomial isomorphism algorithms for the others. Other results in this paper include the development of a structural decomposition for hookup graphs and similar isomorphism results for generalizations of hookup classes, including a polynomial isomorphism testing algorithm for chordal graphs with bounded maximum clique size.

Some perfect coloring properties of graphs

A Grundy n-coloring of a finite graph is a coloring of the points of the graph with the non-negative integers smaller than n such that each point is adjacent to some point of each smaller color but to none of the same color. The Grundy number of a graph is the maximum n for which it has a Grundy n-coloring. Characterizations are given of the families of finite graphs G such that for each induced subgraph H of G: (1) the Grundy number of H is equal to the chromatic number of H; (2) the Grundy number of H is equal to the maximum clique size of H; (3) the achromatic number of H is equal to the chromatic number of H; (4) the achromatic number of H is equal to the maximum clique size of H. The definitions are further extended to infinite graphs, and some of the above characterizations are shown to be true for denumerable graphs and locally finite graphs.