Let G = (V,E) be a graph and let v ∈ V. Let γ(v,G) denote the minimum cardinality of a minimal dominating set of G containing v. Then γ(G) = max{γ(v,G) : v ∈ V (G)} is called the min-max dom-saturation number of G. In this paper we present a dynamic programming algorithm for determining the min-max domsaturation number of a tree.

LetH be a simple graph. A graph G is called anH-saturated graph ifH is not a subgraph of G, but adding any missing edge to Gwill produce a copy of H . Denote by SAT (n,H) the set of all H-saturated graphs G with order n. Then the saturation number sat(n,H) is defined as minG∈SAT (n,H) |E(G)|, and the extremal number ex(n,H) is defined as maxG∈SAT (n,H) |E(G)|. A natural question is that of whet...

A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known ErdősSzekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation p...

Suppose t = (T, T1, p) is a triple of two theories in vocabularies τ ⊂ τ1 with cardinality λ and a τ1-type p over the empty set. We show the Hanf number for the property: ‘There is a model M1 of T1 which omits p, but M1 τ is saturated’ is less than i (2(2 λ)+ ) + if T is superstable. We showed in [BS11] that with no stability restriction the Hanf number is essentially equal to the Löwenheim num...

For fixed graphs $F$ and $H$, a graph $G\subseteq F$ is $H$-saturated if there no copy of $H$ in $G$, but for any edge $e\in E(F)\setminus E(G)$, $G+e$. The saturation number $F$, denoted $sat(F,H)$, the minimum edges an subgraph $F$. In this paper, we study numbers $tK_{l,l,l}$ complete tripartite $K_{n_1,n_2,n_3}$. $t\ge 1$, $l\ge 1$ $n_1,n_2$ $n_3$ sufficiently large, determine $sat(K_{n_1,n...