Unavoidable subgraphs have been widely studied in the context of Ramsey Theory. The research in this area focuses on highly structured graphs such as cliques, cycles, paths, stars, trees, and wheels. We propose to study maximum unavoidable subgraphs measuring the size in the number of edges. We computed maximum unavoidable subgraphs for graphs up to order nine via SAT solving and observed that ...

In the EDGE CLIQUE COVER (ECC) problem, given a graph G and an integer k, we ask whether the edges of G can be covered with k complete subgraphs of G or, equivalently, whether G admits an intersection model on k-element universe. Gramm et al. [JEA 2008] have shown a set of simple rules that reduce the number of vertices of G to 2, and no algorithm is known with significantly better running time...

Partial clique covering is one of the most basic coding schemes for index coding problems, generalizing clique and cycle covering on the side information digraph and further reducing the achievable broadcast rate. In this paper, we start with partition multicast, a special case of partial clique covering with cover number 1, and show that partition multicast achieves the optimal broadcast rate ...

Circle graphs are intersection graphs of chords in a circle and kpolygon graphs are the intersection graphs of chords in a convex ksided polygon where each chord has its endpoints on distinct sides. Every k-polygon graph is a circle graph and every circle graph is a kpolygon graph for some k. The polygon number ψ(G) of a circle graph G is the minimum k such that G is a k-polygon graph and the p...

We introduce a notion of vertex association and consider sequences of these associations. This allows for slick proofs of a few known theorems as well as showing that for any induced subgraph H of G, χ(G) ≤ χ(H) + 12 (ω(G) + |G| − |H| − 1). As a special case of this, we have χ(G) ≤ ⌈ ω(G)+τ(G) 2 ⌉ (here χ(G) denotes the chromatic number, ω(G) the clique number and τ(G) the vertex cover number),...

In cooperative games, a key question is to find a division of payoffs to coalition members in a fair manner. Nucleolus is one of such solution concepts that provides a stable solution for the grand coalition. We study the computation of the nucleolus of a number of cooperative games, including fractional matching games and fractional edge cover games on general weighted graphs, as well as verte...

A minimization (resp., maximization) problem is called fixed parameter (r, t)approximable for two functionsr, t if there exists an algorithm that given an integer k and a problem instance I with optimum value opt, finds either a feasible solution of value at most r(k) · k (resp., at least k/r(k)) or a certificate that k < opt (resp., k > opt), in time t(k) · |I|O(1). A problem is called fixed p...

This thesis is about the design, analysis, implementation, and experimental evaluation of algorithms for hard graph problems. The aim is to establish that the concept of fixed-parameter tractability, and in particular novel algorithmic techniques whose development was driven by this concept, can lead to practically useful programs for exactly solving real-world problem instances. In particular,...

We introduce a novel stochastic local search algorithm for the vertex cover problem. Compared to current exhaustive search techniques, our algorithm achieves excellent performance on a suite of problems drawn from the field of biology. We also evaluate our performance on the commonly used DIMACS benchmarks for the related clique problem, finding that our approach is competitive with the current...

Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph dominates an entire color class. The safe clique partition problem seeks a partition of the vertices of a graph into cliques with the additional property that for each vertex v, there is a clique that has no element in the open neighborhood of v. We typically ...