We study generalizations of the " contraction-deletion " relation of the Tutte polynomial, and other similar simple operations, to other graph parameters. The question can be set in the framework of graph algebras introduced by Freedman, Lovász and Schrijver in [2], and it relates to their behavior under basic graph operations like contraction and subdivision. Graph algebras were introduced in ...

It is known that every graph of sufficiently large chromatic number and bounded clique contains, as an induced subgraph, a subdivision any fixed forest, cycle. Equivalently, forest pervasive, K3 in the class all graphs, where we say H “pervasive” (in some graphs) if for ℓ≥1, has subgraph H, which edge replaced by path at least ℓ edges. Which other graphs are pervasive? was proved Chalopin, Espe...

A polyhedral subdivision of a d-dimensional point configuration A is k-regular if it is projected from the boundary complex of a polytope with dimension at most d+k. Call γk(A) the subgraph induced by k-regular triangulations in the flip-graph of A. Gel’fand, Kapranov, and Zelevinsky have shown that γ1(A) is connected. It is established here that γ2(A) is connected as well. 1. Flip-graph connec...

We study generalizations of the “contraction-deletion” relation of the Tutte polynomial, and other similar simple operations, to other graph parameters. The question can be set in the framework of graph algebras introduced by Freedman, Lovász and Schrijver in [2], and it relates to their behavior under basic graph operations like contraction and subdivision. Graph algebras were introduced in [2...

Hajós conjectured that, for any positive integer k, every graph containing no Kk+1-subdivision is k-colorable. This is true when k ≤ 3, and false when k ≥ 6. Hajós’ conjecture remains open for k = 4, 5. In this paper, we show that any possible counterexample to this conjecture for k = 4 with minimum number of vertices must be 4-connected. This is a step in an attempt to reduce Hajós’ conjecture...

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a 2-dominating set of G if S dominates every vertex of V (G) \ S at least twice. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set of G. The 2-domination subdivision number sdγ2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in ...

We prove that for all positive integers t, every nvertex graph with no Kt-subdivision has at most 2 n cliques. We also prove that asymptotically, such graphs contain at most 2n cliques, where o(1) tends to zero as t tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in n-vertex graphs with no Kt-minor is at most 2 n for some constant c.

In this paper we survey the behavior of various graph operations on the graph parameters clique-width and NLC-width. We give upper and lower bounds for the clique-width and NLC-width of the modi ed graphs in terms of the clique-width and NLC-width of the involved graphs. Therefor we consider the binary graph operations join, co-join, sum, di erence, products, corona, substitution, and 1-sum, an...

In this paper we prove two results. The first is an extension of a result of Dirac which says that any set of n vertices of an n-connected graph lies in a cycle. We prove that if V ′ is a set of at most 2n vertices in an n-connected graph G, then G has, as a minor, a cycle using all of the vertices of V′. The second result says that if G is an n+1-connected graph with maximum vertex degree then...

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of...