Abstract We analyze the problem of computing Banzhaf and Shapley power indices for graph restricted voting games, defined in a particular class graphs, that we called line-clique. A line-clique is model uni-dimensional political space which voters with same bliss point are connected vertices clique then other arcs connect nodes consecutive cliques. The interest to this comes from its correspond...

A graph G=(V,E) is monopolar if V can be partitioned into a stable set and inducing the union of vertex-disjoint cliques. Motivated by an application clique partitioning problem on graphs to cosmetic manufacturing, we study complexity computing classical parameters class graphs. We show that partitioning, stability chromatic numbers NP-hard. Conversely, prove every has polynomial number maximal...

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...

Fox et al. (2020) introduced a new parameter, called c -closure, for parameterized study of clique enumeration problems. A graph G is -closed if every pair vertices with at least common neighbors adjacent. The -closure the smallest such that -closed. We systematically explore impact on computational complexity detecting and enumerating small induced subgraphs. More precisely, each H three or fo...

We introduce and study a constrained planarity testing problem, called 1-Fixed Constrained Planarity, prove that this problem can be solved in quadratic time for biconnected graphs. Our solution is based on novel definition of fixedness makes it possible to simplify extend known techniques about Simultaneous PQ-Ordering. exploit result different versions the hybrid problem. Namely, we show poly...

A graphs G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and is clique reducible if it is not clique irreducible. A graph G is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G and clique vertex reducible if it is not clique vertex irreducible. The clique vertex irred...

Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) problem, Colouring and Minimum Fill-in. The basic tool is a decomposition tree of the...

Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of H...

In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromat...