This paper introduces two kinds of graph polynomials, clique polynomial and independent set polynomial. The paper focuses on expansions of these polynomials. Some open problems are mentioned.

A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel, Lovász and Schrijver 1981). Circular-perfect graphs form a well-studied superclass of perfect graphs. We extend the above result for perfect graphs by showing that clique and chromatic number of a circularperfect graph are computable in polynomial time...

A family of sets is (p,q)-intersecting if every nonempty subfamily p or fewer has at least q elements in its total intersection. the (p,q)-Helly property intersection cardinality q. The (2,1)-Helly usual Helly property. hypergraph edge and hereditary each subhypergraphs graph (p,q)-clique-Helly maximal cliques induced subgraphs (p,q)-clique-Helly. classes (p,q)-biclique-Helly graphs are defined...

A graph has linear clique-width at most k if it has a clique-width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph. We give the first characterisation of graphs of linear clique-width at most 3, and we give the first polynomial-time recognition algorithm for graphs of linear clique-width at most 3. In addition, we present ...

The clique graph K(G) of a graph G is the intersection graph of all its (maximal) cliques, and G is said to be clique divergent if the order of its n-th iterated clique graph Kn(G) tends to infinity with n. In general, deciding whether a graph is clique divergent is not known to be computable. We characterize the dynamical behavior under the clique operator of circulant graphs of the form Cn(a,...

The theoretical information threshold for the planted clique problem is $2\log_2(N)$, however no polynomial algorithm is known to recover a planted clique of size $O(N^{1/2-\epsilon})$, $\epsilon>0$. In this paper we will apply a standard method for the analysis of disordered models, the Parallel-Tempering (PT) algorithm, to the clique problem, showing numerically that its time-scaling in the h...

Berge defined a hypergraph to be balanced if its incidence matrix is balanced. We consider this concept applied to graphs, and call a graph to be balanced when its clique matrix is balanced. Characterizations of balanced graphs by forbidden subgraphs and by clique subgraphs are proved in this work. Using properties of domination we define four subclasses of balanced graphs. Two of them are char...

We provide new tools, such as k-troikas and good subtreerepresentations, that allow us to give fast and simple algorithms computing branchwidth. We show that a graph G has branchwidth at most k if and only if it is a subgraph of a chordal graph in which every maximal clique has a k-troika respecting its minimal separators. Moreover, if G itself is chordal with clique tree T then such a chordal ...

We provide new tools, such as k-troikas and good subtree-representations, that allow us to give fast and simple algorithms computing branchwidth. We show that a graph G has branchwidth at most k if and only if it is a subgraph of a chordal graph in which every maximal clique has a k-troika respecting its minimal separators. Moreover, if G itself is chordal with clique tree T then such a chordal...

Clique separators in graphs are a helpful tool used by Tarjan as a divideand-conquer approach for solving various graph problems such as the Maximum Weight Stable Set (MWS) Problem, Maximum Clique, Graph Coloring and Minimum Fill-in but few examples of graph classes having clique separators are known. We use this method to solve MWS in polynomial time for two classes where the unweighted Maximu...