Graphs without a partition into two proportionally dense subgraphs

A proportionally dense subgraph (PDS) is an induced subgraph of a graph such that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study a partition of a graph into two proportionally dense subgraphs, namely a 2-PDS partition, with and without additional constraint of connectivity of the subgraphs. We present two infinite classes of graphs: one with graphs without a 2-PDS partition, and another with graphs that only admit a disconnected 2-PDS partition. These results answer some questions proposed by Bazgan et al. (2018)

Weighted amplifiers and inapproximability results for Travelling Salesman problem

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The Structure of Obstructions to Treewidth and Pathwidth

It is known that the class of graphs with treewidth (resp. pathwidth) bounded by a constant w can be characterized by a nite obstruction set obs(T W (w)) (resp. obs(PW (w))). These obstruction sets are known for w 3 so far. In this paper we give a structural characterization of graphs from obs(T W (w)) (resp. obs(PW (w))) with a xed number of vertices in terms of subgraphs of the complement. Our approach also essentially simpli es known characterization of graphs from obs(T W (w)) (resp. obs(PW (w))) with (w + 3) vertices

Inapproximability Results for Bounded Variants of Optimization Problems

We study small degree graph problems such as Maximum Independent Set and Minimum Node Cover and improve approximation lower bounds for them and for a number of related problems, like Max-B-Set Packing, Min-B-Set Cover, Max-Matching in B-uniform 2-regular hypergraphs. For example, we prove NP-hardness factor of 94 for Max-3DM, and factor of 47 for Max-4DM; in both cases the hardness result applies even to instances with exactly two occurrences of each element