In this paper a deterministic polynomial-time algorithm is presented for the Clique problem. The case is considered as the problem of omitting the minimum number of vertices from the input graph so that none of the zeroes on the graph’s adjacency matrix (except the main diagonal entries) would remain on the adjacency matrix of the resulting subgraph. The existence of a deterministic polynomial-...

Cardinality Maximum Flow Network Interdiction Problem (CMFNIP) is known to be strongly NP-hard problem in the literature. A particular case of CMFNIP has been shown to have reduction from clique problem. In the present work,an effort is being made to solve this particular case of CMFNIP in polynomial time. Direct implication of this solution is that the clique problem gets solved in polynomial ...

We show that 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 polynomial time. We show that the independent set problem on edge-clique ...

Clique-width is an important graph parameter whose computation is NP-hard. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any non-trivial graph class. Results so far indicate that proper interval graphs constitute the first interesting graph class on which we might have hope to compute clique-width, or at least its linear variant line...

Numerous problems can be modeled as clique partitioning problems in digital design synthesis. In this paper, we present two new polynomial time heuristic algorithms for efficient clique partitioning with or without limiting the maximum clique size. The goal of clique partitioning is to partition a graph into a minimum number of cliques. The basic approach of the new algorithm is to find small c...

Maximum Flow Network Interdiction Problem (MFNIP) is known to be strongly NP-hard problem. We solve a simple form of MFNIP in polynomial time. We review the reduction of MFNIP from the clique problem. We propose a polynomial time solution to the Clique Problem.

In this work, we critique two papers, “A Polynomial-Time Solution to the Clique Problem” by Tamta, Pande, and Dhami [3], and “A Polynomial-Time Algorithm For Solving Clique Problems” by LaPlante [1]. We summarize and analyze both papers, noting that the algorithms presented in both papers are flawed. We conclude that neither author has successfully established that P = NP.

The chromatic polynomial gives the number of proper λ-colourings of a graph G. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if P (G,λ) = P (H1, λ)P (H2, λ)/P (Kr , λ) for some graphs H1 and H2 and clique Kr. It is known that the c...

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 such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph canno...

A k-path power is the k-power graph of a simple path of arbitrary length. Path powers form a non-trivial subclass of proper interval graphs. Their clique-width is not bounded by a constant, and no polynomial-time algorithm is known for computing their clique-width or linear clique-width. We show that k-path powers above a certain size have linear clique-width exactly k + 2, providing the first ...