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 length at most k which neither starts nor ends in U. Identifying a k-kernel of a graph G enables one to construct an infinite family of graphs from G which have depression at most k. We discuss various results related to the concept of k-kernels, including an improved upper bound for the depression of trees.
@article{bwmeta1.element.doi-10_7151_dmgt_1736,
author = {Mark Schurch and Christine Mynhardt},
title = {The depression of a graph and k-kernels},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {233-247},
zbl = {1290.05128},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1736}
}
Mark Schurch; Christine Mynhardt. The depression of a graph and k-kernels. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 233-247. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1736/
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