Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this work we consider parameterized complexity problem. We show proper/unit graphs FPT when by number non-empty bubbles a column its bubble model. then generalize result more general graph class defining...

We extend the work of Letchford (2000) by introducing a new class of valid inequalities for the traveling salesman problem, called the generalized domino-parity (GDP) constraints. Just as Letchford’s domino-parity constraints generalize comb inequalities, GDP constraints generalize the most well-known multiple-handle constraints, including clique-tree, bipartition, path, and star inequalities. ...

We design an O(m) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n’) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordere...

We have seen that in any graph G = (V, E), a maximum-size matching can be found in polynomial time. This means that α(L(G)) can be found in polynomial time, where L(G) is the line graph of G.2 On the other hand, it is NP-complete to find a maximum-size stable set in a graph. That is, determining α(G) is NP-complete. Since α(G) = |V |−τ(G) and α(G) = ω(G), also determining the vertex cover numbe...

We have seen that in any graph G = (V,E), a maximum-size matching can be found in polynomial time. This means that α(L(G)) can be found in polynomial time, where L(G) is the line graph of G.2 On the other hand, it is NP-complete to find a maximum-size stable set in a graph. That is, determining α(G) is NP-complete. Since α(G) = |V |−τ(G) and α(G) = ω(G), also determining the vertex cover number...

A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grötschel, Lovász, and Schrijver [9] from 1988 finds an optimal colouring of a perfect graph in polynomial time. But this algorithm uses the ellipsoid method, and it is a well-known open question to construct a “combinatorial” polynomial-time algorithm that yields an o...

A graph complexity measure that we call clique-width is associated in a natural way with certain graph decompositions, more or less like tree-width is associated with tree-decomposition which are, actually, hierarchical decompositions of graphs. In general, a decomposition of a graph G can be viewed as a nite term, written with appropriate operations on graphs, that evaluates to G. Innnitely ma...

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel by proving a GapL ⊆ NC ⊆ P upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded clique-width graphs can be performed efficiently in parallel. Thus we show...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...