The (p)-neighborhood graph of a graph G, denoted N p (G), is de-ned on the same vertex set as G, with x; y] 2 E (N 2 (G)) if and only if jN(x)\N(y)j p in G, where N (v) is the open neighborhood of vertex v. The p]-neighborhood graph of G, N p G], is deened similarly, using closed neighborhoods rather than open ones. If G is the underlying graph of a symmetric digraph D, then the p-neighborhood ...

The class of acyclic clique-interval (ACI) graphs is introduced as the class of those graphs G=(V ,E) whose cliques are intervals (chains) of an acyclic order on the vertex set V . The class of ACI graphs is related to the classes of proper interval graphs, tree-clique graphs and to the class DV (intersection graphs of directed paths of a directed tree). Compatibility between a graph and an acy...

Gavril [GA4] defined two new families of intersection graphs: the interval-filament graphs and the subtree-filament graphs. The complements of intervalfilament graphs are the cointerval mixed graphs and the complements of subtree-filament graphs are the cochordal mixed graphs. The family of interval-filament graphs contains the families of cocomparability, polygon-circle, circle and chordal gra...

Let x(G) and w(G) denote the chromatic number and clique number (maximum size of a clique) of a graph G. To avoid trivial cases, we always assume that w (G);?: 2. It is well known that interval graphs are perfect, in particular x( G)= w (G) for every interval graph G. In this paper we study the closeness of x and w for two well-known non-perfect relatives of interval graphs: multiple interval g...

A d-track interval is a union of d intervals, one each from d parallel lines. The intersection graphs of d-track intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The track number for K m;n is determined by proving that the arboricity of K m;n equals its \caterpillar arbori...

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph ...

We consider models for random interval graphs that are based on stochastic service systems, with vertices corresponding to customers and edges corresponding to pairs of customers that are in the system simultaneously. The number N of vertices in a connected component thus corresponds to the number of customers arriving during a busy period, while the size K of the largest clique (which for inte...

Let ! be the induced-minor relation. It is shown that, for every t, all chordal graphs of clique number at most t are well-quasi-ordered by !. On the other hand, if the bound on clique number is dropped, even the class of interval graphs is not well-quasi-ordered by !. c © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 105–114, 1998

Let G be a connected simple graph. We prove that G is a closed graph if and only if G is a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.