The Clique Problem has a reduction to the Maximum Flow Network Interdiction Problem. We review the reduction to evolve a polynomial time algorithm for the Clique Problem. A computer program in C language has been written to validate the easiness of the algorithm.

Clique-width is a graph parameter, defined by a composition mechanism for vertexlabeled graphs, which measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic, that includes NPhard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a...

This paper describes the design of a parallel algorithm that uses moving fluids in a three-dimensional microfluidic system to solve a nondeterministically polynomial complete problem (the maximal clique problem) in polynomial time. This algorithm relies on (i) parallel fabrication of the microfluidic system, (ii) parallel searching of all potential solutions by using fluid flow, and (iii) paral...

A class G of graphs is said to be χ-bounded if there is a function f : N → R such that for all G ∈ G and all induced subgraphs H of G, χ(H) ≤ f(ω(H)). In this paper, we show that if G is a χ-bounded class, then so is the closure of G under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if G is χ-bounded...

In a directed graph, a kernel is a subset of the vertices that is both independent and absorbing. Not all directed graphs have a kernel, and finding classes of graphs having always a kernel or for which deciding the existence of a kernel is polynomial has been the topic of many works in graph theory. We formalize some techniques to build a kernel in a graph with a clique-cutset, knowing kernels...

We introduce a graph transformation which preserves the clique number. When applied to graphs containing no odd hole and no cricket (a particular graph on 5 vertices) the transformation also preserves the chromatic number. Using this transformation we derive a polynomial algorithm for the computation of the clique number of all graphs in a class which strictly contains diamond-free graphs. Furt...

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). This result relies on polyhedral characterizations of perfect graphs involving the stable set polytope of the graph, a linear relaxation defined by clique constraints, and a semi-definite relaxation, the Theta-body of the gr...

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-h...

We offer an algorithm that finds a clique tree such that the size of the largest clique is at most (2α + 1)k where k is the size of the largest clique in a clique tree in which this size is minimized and α is the approximation ratio of an α-approximation algorithm for the 3-way vertex cut problem. When α = 4/3, our algorithm’s complexity is O(24.67kn · poly(n)) and it errs by a factor of 3.67 w...

If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M...