# The depression of a graph and k-kernels

• Volume: 34, Issue: 2, page 233-247
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Mark Schurch, and Christine Mynhardt. "The depression of a graph and k-kernels." Discussiones Mathematicae Graph Theory 34.2 (2014): 233-247. <http://eudml.org/doc/267588>.

@article{MarkSchurch2014,
abstract = {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.},
author = {Mark Schurch, Christine Mynhardt},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {edge ordering of a graph; increasing path; monotone path; de- pression; depression},
language = {eng},
number = {2},
pages = {233-247},
title = {The depression of a graph and k-kernels},
url = {http://eudml.org/doc/267588},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Mark Schurch
AU - Christine Mynhardt
TI - The depression of a graph and k-kernels
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 2
SP - 233
EP - 247
AB - 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.
LA - eng
KW - edge ordering of a graph; increasing path; monotone path; de- pression; depression
UR - http://eudml.org/doc/267588
ER -

## References

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