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Convex universal fixers

Magdalena LemańskaRita Zuazua — 2012

Discussiones Mathematicae Graph Theory

In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and E ( π G ) = E ( G ) E ( G ' ) M π , where M π = u π ( u ) | u V ( G ) . Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept of universal...

Self-diclique circulant digraphs

Marietjie FrickBernardo LlanoRita Zuazua — 2015

Mathematica Bohemica

We study a particular digraph dynamical system, the so called digraph diclique operator. Dicliques have frequently appeared in the literature the last years in connection with the construction and analysis of different types of networks, for instance biochemical, neural, ecological, sociological and computer networks among others. Let D = ( V , A ) be a reflexive digraph (or network). Consider X and Y (not necessarily disjoint) nonempty subsets of vertices (or nodes) of D . A disimplex K ( X , Y ) of D is the subdigraph...

Total Domination Multisubdivision Number of a Graph

Diana Avella-AlaminosMagda DettlaffMagdalena LemańskaRita Zuazua — 2015

Discussiones Mathematicae Graph Theory

The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdi- vision number is equal to the known total domination subdivision...

Distance 2-Domination in Prisms of Graphs

Ferran HurtadoMercè MoraEduardo Rivera-CampoRita Zuazua — 2017

Discussiones Mathematicae Graph Theory

A set of vertices D of a graph G is a distance 2-dominating set of G if the distance between each vertex u ∊ (V (G) − D) and D is at most two. Let γ2(G) denote the size of a smallest distance 2-dominating set of G. For any permutation π of the vertex set of G, the prism of G with respect to π is the graph πG obtained from G and a copy G′ of G by joining u ∊ V(G) with v′ ∊ V(G′) if and only if v′ = π(u). If γ2(πG) = γ2(G) for any permutation π of V(G), then G is called a universal γ2-fixer. In this...

Some Variations of Perfect Graphs

Magda DettlaffMagdalena LemańskaGabriel SemanišinRita Zuazua — 2016

Discussiones Mathematicae Graph Theory

We consider (ψk−γk−1)-perfect graphs, i.e., graphs G for which ψk(H) = γk−1(H) for any induced subgraph H of G, where ψk and γk−1 are the k-path vertex cover number and the distance (k − 1)-domination number, respectively. We study (ψk−γk−1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (ψ2 − γ1)- perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging...

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