A nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k \rangle$ is said to be {\em $n$-realizable\/} if the set $I_n=\{ 1,2,\cdots,n\}$ can be partitioned into $k$ mutually disjoint subsets $S_1,S_2,\cdots, S_k$ such that $\sum\limits_{x\in S_i}x=m_i$ for each $1\le i\le k$. In this paper, we will prove that a nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k\rangle$ is … Read more
An operation on trivalent graphs leads from the truncated cube to buckminsterfullerene, and C60 is the only fullerene with disjoint pentagons which can be obtained by this method. The construction and the proof emphasize maximal independent sets that contain two fifths of the vertices of trivalent graphs. In the case of C60, these sets define … Read more
We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lov\’asz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures’ performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations … Read more
Taking a small graph, on which the randomized New-Best-In maximum clique heuristic fails to find the maximum clique, we construct on its basis a class of graphs exemplifying the inefficiency of SM greedy heuristics considered by Brockington and Culberson. We show that a 7(k+1)-vertex graph from this class is enough to provide a counterexample for … Read more
Arranging an n-vertex graph as the standard simplex in R^n, we identify graph cliques with simplex faces formed by clique vertices. An unstrict quadratic inequality holds for all points of the simplex; it turns to equality if and only if the point is on a face corresponding to a clique. This way this equality determines … Read more
Since the Motzkin-Straus result on the clique number of graphs, published in 1965, where they show that the size of the largest clique in a graph can be obtained by solving a quadratic programming problem, several results on the continuous approach to the determination of the clique number of a graph or, equivalently, to the … Read more
In one of fundamental work in combinatorial optimization Edmonds gave a complete linear description of the matching polytope. Matchings in a graph are equivalent to stable sets its line graph. Also the neighborhood of any vertex in a line graph partitions into two cliques: graphs with this latter property are called quasi-line graphs. Quasi-line graphs … Read more
We introduce a graph invariant, called thinness, and show that a maximum weighted stable set on a graph $G(V, E)$ with thinness $k$ may be found in $O(\frac{|V|}{k})^k$-time, if a certain representation is given. We show that a graph has thinness 1 if and only if it is an interval graph, while a graph with … Read more
We describe a new neighborhood structure for the capacitated minimum spanning tree problem. This neighborhood structure is used by a local search strategy, leading to good trade-offs between solution quality and computation time. We also propose a GRASP with path-relinking heuristic. It uses a randomized version of a savings heuristic in the construction phase and … Read more
Many NP-hard graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for many optimization … Read more