Line-graphs of cubic graphs are normal

A graph is called normal if its vertex set can be covered by cliques and also by stable sets, such that every such clique and stable set have non-empty intersection. This notion is due to Korner, who introduced the class of normal graphs as an extension of the class of perfect graphs. Normality has also relevance in information theory. Here we prove, that the line graphs of cubic graphs are normal.Comment: 16 pages, 10 figure

Satgraphs and independent domination. Part 1

AbstractA graph G is called a satgraph if there exists a partition A∪B=V(G) such that•A induces a clique [possibly, A=∅],•B induces a matching [i.e., G(B) is a 1-regular subgraph, possibly, B=∅], and•there are no triangles (a,b,b′), where a∈A and b,b′∈B.We also introduce the hereditary closure of SAT, denoted by HSAT [hereditary satgraphs]. The class HSAT contains split graphs. In turn, HSAT is contained in the class of all (1,2)-split graphs [A. Gyárfás, Generalized split graphs and Ramsey numbers, J. Combin. Theory Ser. A 81 (2) (1998) 255–261], the latter being still not characterized. We characterize satgraphs in terms of forbidden induced subgraphs.There exist close connections between satgraphs and the satisfiability problem [SAT]. In fact, SAT is linear-time equivalent to finding the independent domination number in the corresponding satgraph. It follows that the independent domination problem is NP-complete for the hereditary satgraphs. In particular, it is NP-complete for perfect graphs

Independent domination in hereditary classes

AbstractWe investigate Independent Domination Problem within hereditary classes of graphs. Boliac and Lozin [Independent domination in finitely defined classes of graphs, Theoret. Comput. Sci. 301 (1–3) (2003) 271–284] proved some sufficient conditions for Independent Domination Problem to be NP-complete within finitely defined hereditary classes of graphs. They posed a question whether the conditions are also necessary. We show that the conditions are not necessary, since Independent Domination Problem is NP-hard within 2P3-free graphs.Moreover, we show that the problem remains NP-hard for a new hereditary class of graphs, called hereditary 3-satgraphs. We characterize hereditary 3-satgraphs in terms of forbidden induced subgraph. As corollaries, we prove that Independent Domination Problem is NP-hard within the class of all 2P3-free perfect graphs and for K1,5-free weakly chordal graphs.Finally, we compare complexity of Independent Domination Problem with that of Independent Set Problem for a hierarchy of hereditary classes recently proposed by Hammer and Zverovich [Construction of maximal stable sets with k-extensions, Combin. Probab. Comput. 13 (2004) 1–8]. For each class in the hierarchy, a maximum independent set can be found in polynomial time, and the hierarchy covers all graphs. However, our characterization of hereditary 3-satgraphs implies that Independent Domination Problem is NP-hard for almost all classes in the hierarchy. This fact supports a conjecture that Independent Domination is harder than Independent Set Problem within hereditary classes

Voting in Agreeable Societies

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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

All Minimal Prime Extensions of Hereditary Classes of Graphs

The substitution composition of two disjoint graphs G1 and G2 is obtained by first removing a vertex x from G2 and then making every vertex in G1 adjacent to all neighbours of x in G2. Let F be a family of graphs defined by a set Z* of forbidden configurations. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83–97] proved that F∗, the closure under substitution of F, can be characterized by a set Z∗ of forbidden configurations — the minimal prime extensions of Z. He also showed that Z∗ is not necessarily a finite set. Since substitution preserves many of the properties of the composed graphs, an important problem is the following: find necessary and sufficient conditions for the finiteness of Z∗. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83–97] presented a sufficient condition for the finiteness of Z∗ and a simple method for enumerating all its elements. Since then, many other researchers have studied various classes of graphs for which the substitution closure can be characterized by a finite set of forbidden configurations. The main contribution of this paper is to completely solve the above problem by characterizing all classes of graphs having a finite number of minimal prime extensions. We then go on to point out a simple way for generating an infinite number of minimal prime extensions for all the other classes of F∗