Let G = (V,E) be a directed/undirected graph, let s, t ∈ V , and let F be an intersecting family on V (that is, X ∩ Y,X ∪ Y ∈ F for any intersecting X,Y ∈ F) so that s ∈ X and t / ∈ X for every X ∈ F . An edge set I ⊆ E is an edge-cover of F if for every X ∈ F there is an edge in I from X to V − X. We show that minimal edge-covers of F can be listed with polynomial delay, provided that for any ...

In an attempt to find a polynomial-time algorithm for the edgeclique cover problem on cographs we tried to prove that the edge-clique graphs of cographs have bounded rankwidth. However, this is not the case. In this note we show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth.

K.Dohmen, A.Pönitz and P.Tittman (2003), introduced a bivariate generalization of the chromatic polynomial P (G, x, y) which subsumes also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We first show that P (G, x, y) has a recursive definition with respect to three kinds of edge elimi...

The tree and tour cover problems on an edge-weighted graph are to compute a minimum weight tree and closed walk, respectively, whose vertices form a vertex cover. Both problems are NP-hard. In this note we give strongly polynomial time, constant factor approximation algorithms for both problems. An interesting feature of our algorithms is how they combine approximations of other problems, namel...

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and edge extraction, i.e., deletion of edges together with their end points. Like in the case of deletion and contraction only (J.G. Oxley and D.J.A. Welsh 1979), it turns out that there is a most general polynomial satisfying such recurrence relations, which we c...

The construction of probabilistic models that can represent large systems requires the ability to describe repetitive and hierarchical structures. To do so, one can resort to constructs from description logics. In this paper we present a class of relational Bayesian networks based on the popular description logic DL-Lite. Our main result is that, for this modeling language, marginal inference a...

We present approximation algorithms for four variations of the maximum latency problem. We consider symmetric graphs and asymmetric graphs and both with general edge weights or weights satisfying the triangle inequality. Moreover, in each variation the starting point of the tour may either be given in the input or be a decision variable. As a tool for our solution, we use a PTAS for the maximum...

The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how fast this “deterministic random walk” covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several impo...

A set P of edge-disjoint paths in a graph G is called an edge-covering of G if every edge of G is contained in a path of P. The path edge-covering number of G is then defined as the smallest number p(G) of paths in such an edge-covering of G. This parameter appeared in the study of certain models of information retrieval structures (see, e. g., [6]). If we are considering a tree T , then it is ...

We use a new approximation measure, the differential approximation ratio, to derive polynomial-time approximation algorithms for minimum set covering (for both weighted and unweighted cases), minimum graph coloring and bin-packing. We also propose differentialapproximation-ratio preserving reductions linking minimum coloring, minimum vertex covering by cliques, minimum edge covering by cliques ...