Families of all sets of independent vertices in graphs are investigated. The problem how to characterize those infinite graphs which have arithmetically maximal independent sets is posed. A positive answer is given to the following classes of infinite graphs: bipartite graphs, line graphs and graphs having locally infinite clique-cover of vertices. Some counter examples are presented.

Let T = (V,A) be a directed tree. Given a collection P of dipaths on T , we can look at the arc-intersection graph I(P, T ) whose vertex set is P and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs)...

The clique cover number θ1(G) of a graph G is the minimum number of cliques required to cover the edges of graph G. In this paper we consider θ1(Gn,p), for p constant. (Recall that in the random graph Gn,p, each of the ( n 2 ) edges occurs independently with probability p). Bollobás, Erdős, Spencer and West [1] proved that whp (i.e. with probability 1-o(1) as n→∞) (1− o(1))n 4(log2 n) 2 ≤ θ1(Gn...

Surface meltwater can refreeze within firn layers and crevasses to warm ice through latentheat transfer on decadal to millennial timescales. Earlier work posited that the consequent softening of the ice might accelerate ice flow, potentially increasing ice-sheet mass loss. Here, we calculate the effect of meltwater refreezing on ice temperature and softness in the Pâkitsoq (near Swiss Camp) and...

We describe and analyze test case generators for the maximum clique problem (or equivalently for the maximum independent set or vertex cover problems). The generators produce graphs with speci ed number of vertices and edges, and known maximum clique size. The experimental hardness of the test cases is evaluated in relation to several heuristics for the maximum clique problem, based on neural n...

Many introductions to the theory of NP-completeness make some mention of approximating NP-complete problems. The usual story line says that even though solving an NP-complete problem exactly may be intractable, it is sometimes possible to find an approximate solution in polynomial time. The intuitive assumption is that finding an approximate solution should be easier than finding the exact solu...

Computing the clique number and chromatic number of a general graph are well-known to be NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl., 285(1-3): 123–142, 1998) showed that computing the clique number and chromatic number are still NP-Hard probl...