On local structures of cubicity 2 graphs

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation

Deciding the Bell Number for Hereditary Graph Properties

The paper [J. Balogh, B. Bollobás, D. Weinreich, J. Combin. Theory Ser. B, 95 (2005), pp. 29--48] identifies a jump in the speed of hereditary graph properties to the Bell number $B_n$ and provides a partial characterization of the family of minimal classes whose speed is at least $B_n$. In the present paper, we give a complete characterization of this family. Since this family is infinite, the decidability of the problem of determining if the speed of a hereditary property is above or below the Bell number is questionable. We answer this question positively by showing that there exists an algorithm which, given a finite set $\mathcal{F}$ of graphs, decides whether the speed of the class of graphs containing no induced subgraphs from the set $\mathcal{F}$ is above or below the Bell number. For properties defined by infinitely many minimal forbidden induced subgraphs, the speed is known to be above the Bell number. Read More: http://epubs.siam.org/doi/abs/10.1137/15M102421

The combinatorics of resource sharing

We discuss general models of resource-sharing computations, with emphasis on the combinatorial structures and concepts that underlie the various deadlock models that have been proposed, the design of algorithms and deadlock-handling policies, and concurrency issues. These structures are mostly graph-theoretic in nature, or partially ordered sets for the establishment of priorities among processes and acquisition orders on resources. We also discuss graph-coloring concepts as they relate to resource sharing.Comment: R. Correa et alii (eds.), Models for Parallel and Distributed Computation, pp. 27-52. Kluwer Academic Publishers, Dordrecht, The Netherlands, 200

Clique covering the edges of a locally cobipartite graph

AbstractWe prove that a locally cobipartite graph on n vertices contains a family of at most n cliques that cover its edges. This is related to Opsut's conjecture that states the competition number of a locally cobipartite graph is at most two