A Sum of Squares Characterization of Perfect Graphs

We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sums of squares. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sums of squares through graph-theoretic constructions. We also characterize graphs for which the associated polynomials belong to certain structured subsets of sum of squares polynomials. Finally, we reformulate some well-known results from the theory of perfect graphs as statements about sum of squares proofs of nonnegativity of certain polynomials

Graphs with polynomially many minimal separators

We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are excluded. As a consequence, there is a polynomial time algorithm to solve the maximum weight independent set problem for the class of (theta, pyramid, prism, turtle)-free graphs. Since every prism, theta, and turtle contains an even hole, this also implies a polynomial time algorithm to solve the maximum weight independent set problem for the class of (pyramid, even hole)-free graphs

Perfect Graphs and Sums of Squares

A graph is perfect if the clique number and the chromatic number coincide for any of its induced subgraphs. A polynomial is a sum of squares (sos) if it can be written as a sum of squares of some other polynomials. In the first technical chapter of this thesis, we bring these two notions together by presenting an algebraic characterization of perfect graphs. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sos. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sos through graph-theoretic constructions. We also establish a number of other connections between structural graph theory and real algebra. In the subsequent chapters, we focus on problems that are relevant to sos polynomials and perfect graphs individually. In one chapter, we study separable plus quadratic (SPQ) polynomials. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sos, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial, and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials, but negative already for bivariate quartic SPQ polynomials. For question (ii), we provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. We end the chapter by presenting applications of SPQ polynomials to problems in statistics and nonlinear optimization. In the last two chapters, we focus on the study of graphs that are closely related to the class of perfect graphs, namely even-hole-free graphs and strongly perfect graphs. Although the class of even-hole-free graphs is a widely studied class of graphs, the complexity of the maximum independent set problem in this class is a long-standing open problem. We take a step forward in this direction by showing that there is a polynomial-time algorithm to solve the maximum independent set problem in the class of (pyramid, even hole)-free graphs. Regarding strongly perfect graphs, although their characterization by a set of forbidden induced subgraphs remains open, we provide several new infinite families of minimal non-strongly-perfect graphs. We also present a new proof of the characterization of claw-free strongly perfect graphs, which is shorter and quite different from the original proof

On matching extendability of lexicographic products

A graph G of even order is ℓ-extendable if it is of order at least 2ℓ + 2, contains a matching of size ℓ, and if every such matching is contained in a perfect matching of G. In this paper, we study the extendability of lexicographic products of graphs. We characterize graphs G and H such that their lexicographic product is not 1-extendable. We also provide several conditions on the graphs G and H under which their lexicographic product is 2-extendable