Optimal Partition Trees

Global Alliance Partition in Trees

Let G be a connected graph with vertex set V (G) and edge set E(G). A (defensive) alliance in G is a subset S of V (G) such that for every vertex v ∈ S, |N [v]∩S| ≥ |N(v)∩(V (G)− S)|. The alliance partition number, ψa(G), was defined (and further studied in [11]) to be the maximum number of sets in a partition of V (G) such that each set is a (defensive) alliance. Similarly, ψg(G) is the maximu...

The Optimal Cost Chromatic Partition Problem for Trees and Interval Graphs

In this paper we study the Optimal Cost Chromatic Partition (OCCP) problem for trees and interval graphs. The OCCP problem is the problem of coloring the nodes of a graph in such a way that adjacent nodes obtain diierent colors and that the total coloring costs are minimum. In this paper we rst give a linear time algorithm for the OCCP problem for trees. The OCCP problem for interval graphs is ...

Optimal Trees

On the partition dimension of trees

Given an ordered partition Π = {P1, P2, ..., Pt} of the vertex set V of a connected graph G = (V,E), the partition representation of a vertex v ∈ V with respect to the partition Π is the vector r(v|Π) = (d(v, P1), d(v, P2), ..., d(v, Pt)), where d(v, Pi) represents the distance between the vertex v and the set Pi. A partition Π of V is a resolving partition of G if different vertices of G have ...

A canonical partition theorem for trees

We show that for every positive integer d, every downwards closed subtree T of d without terminal nodes and every finite weakly embedded subtree A of T there is a finite list of equivalence relations on Em(T ) with the property that for every other equivalence relation E on Em(T ) there is a strongly embedded subtree S ⊆ T of height ω, such that E Em(S) is equal to one of the equivalence relati...