A graph is (P5,gem)-free, when it does not contain P5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P4) as an induced subgraph. We present O(n2) time recognition algorithms for chordal gem-free and for (P5,gem)free graphs. Using a characterization of (P5,gem)-free graphs by their prime graphs...

Fix integers n ≥ r ≥ 2. A clique partition of ( [n] r ) is a collection of proper subsets A1, A2, . . . , At ⊂ [n] such that ⋃ i ( Ai r ) is a partition of ( [n] r ) . Let cp(n, r) denote the minimum size of a clique partition of ( [n] r ) . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, cp(n, ...

We study projective dimension, a graph parameter (denoted by pd(G) for a graph G), introduced by Pudlák and Rödl [13], who showed that proving lower bounds for pd(Gf ) for bipartite graphs Gf associated with a Boolean function f imply size lower bounds for branching programs computing f . Despite several attempts [13, 17], proving super-linear lower bounds for projective dimension of explicit f...

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph ...

Coded caching is an effective technique to reduce the redundant traffic in wireless networks. The existing coded caching schemes require the splitting of files into a possibly large number of subfiles, i.e., they perform coded subfile caching. Keeping the files intact during the caching process would actually be appealing, broadly speaking because of its simpler implementation. However, little ...

We will remove vertices from a given graph G of treewidth k in a good elimination ordering, and we can assume w.l.o.g. that G is connected. Some cliques in a current graph will be active cliques. For each active clique C and for each subset S of vertices in C, we store information if there exists a vertex cover of the original graph G that chooses exactly vertices in S out of the vertices in C....

The vertex-disjoint triangles (VDT) problem asks for a set of maximum number of pairwise vertex-disjoint triangles in a given graph G. The triangle cover problem asks for the existence of a perfect triangle packing in a graph G. It is known that the triangle cover problem is NPcomplete on general graphs with clique number 3 [6]. The VDT problem is MAX SNP-hard on graphs with maximum degree four...

We show that for a graph G it is NP-hard to decide whether its independence number (G) equals its clique partition number (G) even when some minimum clique partition of G is given. This implies that any (G)-upper bound provably better than (G) is NPhard to compute. To establish this result we use a reduction of the quasigroup completion problem (QCP, known to be NP-complete) to the maximum inde...

We investigate the problem of coded caching for nonuniform demands when the structured clique cover algorithm proposed by Maddah-Ali and Niesen for decentralized caching is used for delivery. We apply this algorithm to all user demands regardless of their request probabilities. This allows for coding among the files that have different request probabilities but makes the allocation of memory to...

The minimum vector rank (mvr) of a graph over a field F is the smallest d for which a faithful vector representation of G exists in Fd . For simple graphs, minimum semidefinite rank (msr) and minimum vector rank differ by exactly the number of isolated vertices. We explore the relationship between msr and mvr for multigraphs and show that a result linking the msr of chordal graphs to clique cov...