The matching preclusion number of a graph is the minimum number of neither edges whose deletion in a graph has a neither perfect matching nor an almost perfect matching. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently the conditional matching preclusion number of a graph was introduced to look for sets beyond those induced by a single...

There is perfect state transfer between two vertices of a graph, if a single excitation can travel with fidelity one between the corresponding sites of a spin system modeled by the graph. When the excitation is back at the initial site, for all sites at the same time, the graph is said to be periodic. A graph is cubic if each of its vertices has a neighbourhood of size exactly three. We prove t...

In the Perfect Completion problem one wishes to add the fewest possible edges to a graph in order to obtain a perfect graph. How large can the size of the added edge set be compared to the size of the edge set of the original graph? We show that with high probability the smallest perfect graph containing a random graph G(n; n 0:6) has (n 1:8) edges. We also show that with high probability the m...

It is an open question whether the chromatic number of t-perfect graphs is bounded by a constant. The largest known value for this parameter is 4, and the only example of a 4-critical t-perfect graph, due to Laurent and Seymour, is the complement of the line graph of the prism Π (a graph is 4-critical if it has chromatic number 4 and all its proper induced subgraphs are 3-colorable). In this pa...

A perfect graph is critical if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We relate critically perfect graphs to well-known classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness.

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the exten...

A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear progr...