This work examines the problem of clique enumeration on a graph by exploiting its covers. The principle inclusion/exclusion is applied to determine number cliques size $r$ in union set $\mathcal{C} = \{c_1, \ldots, c_m\}$ $m$ cliques. leads deeper examination sets involved and an orbit partition, $\Gamma$, power $\mathcal{P}(\mathcal{N}_{m})$ $\mathcal{N}_{m} \{1, m\}$. Applied cliques, this pa...

The sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in Rd . The edge clique cover number (G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G) (G). Our upper bound remains valid for intersection graphs defined by bal...

The linear vertex-arboricity of a graph G is defined to the minimum number of subsets into which the vertex-set G can be partitioned so that every subset induces a linear forest. In this paper, we give the upper and lower bounds for sum and product of linear vertex-arboricity with independence number and with clique cover number respectively. All of these bounds are sharp.

Given two graphs G and H, assume that C = {C1, C2, . . . , Cq} is a clique cover of G and U is a subset of V (H). We introduce a new graph operation called the clique cover product, denoted by G ⋆ HU , as follows: for each clique Ci ∈ C , add a copy of the graph H and join every vertex of Ci to every vertex of U . We prove that the independence polynomial of G ⋆ HU I(G ⋆ H ;x) = I(H;x)I(G; xI(H...

A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs some problems that are NP-complete on general graphs, such as the maximum...

We consider the problem of edge clique cover on sparse networks and study an application to the identification of overlapping protein complexes for a network of binary protein-protein interactions. We first give an algorithm whose running time is linear in the size of the graph, provided the treewidth is bounded. We then provide an algorithm for planar graphs with bounded branchwidth upon which...

For a clique cover C in the undirected graph G, the clique cover graph of C is the graph obtained by contracting the vertices of each clique in C into a single vertex. The clique cover width of G, denoted by CCW (G), is the minimum value of the bandwidth of all clique cover graphs in G. Any G with CCW (G) = 1 is known to be an incomparability graph, and hence is called, a unit incomparability g...

A γ-quasi-clique in a simple undirected graph is a set of vertices which induces a subgraph with the edge density of at least γ for 0 < γ < 1. A cover of a graph by γ-quasi-cliques is a set of γ-quasi-cliques where each edge of the graph is contained in at least one quasi-clique. The minimum cover by γ-quasi-cliques problem asks for a γ-quasi-clique cover with the minimum number of quasi-clique...

We show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth. This, and other observations lead us to conjecture that the edge-clique cover problem is NP-complete for cographs. We show that the independent set problem on edge-clique graphs of cographs and of distance-hereditary graphs can be solved in O(n4) time. We show that the independent set problem on edge-clique g...

Assume that G = G(V, E) is an undirected graph with vertex set V and edge set E. A clique of G is a complete subgraph. An edge clique-covering is a family of cliques of G which cover all edges of G. The edge clique-cover number, Be( G), is the minimum number of cliques in an edge clique-cover of G. For results and applications of the edge clique-cover number see [1-4]. Observe that Be(G) does n...