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The depression of a graph and k-kernels

Mark SchurchChristine Mynhardt — 2014

Discussiones Mathematicae Graph Theory

An edge ordering of a graph G is an injection f : E(G) → R, the set of real numbers. A path in G for which the edge ordering f increases along its edge sequence is called an f-ascent ; an f-ascent is maximal if it is not contained in a longer f-ascent. The depression of G is the smallest integer k such that any edge ordering f has a maximal f-ascent of length at most k. A k-kernel of a graph G is a set of vertices U ⊆ V (G) such that for any edge ordering f of G there exists a maximal f-ascent of...

Paired domination in prisms of graphs

Christina M. MynhardtMark Schurch — 2011

Discussiones Mathematicae Graph Theory

The paired domination number γ p r ( G ) of a graph G is the smallest cardinality of a dominating set S of G such that ⟨S⟩ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: γ p r ( π G ) = 2 γ p r ( G ) for all πG; γ p r ( K G ) = 2 γ p r ( G ) ; γ p r ( K G ) = γ p r ( G ) .

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