865 research outputs found
Induced minors and well-quasi-ordering
A graph is an induced minor of a graph if it can be obtained from an
induced subgraph of by contracting edges. Otherwise, is said to be
-induced minor-free. Robin Thomas showed that -induced minor-free
graphs are well-quasi-ordered by induced minors [Graphs without and
well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 --
247, 1985].
We provide a dichotomy theorem for -induced minor-free graphs and show
that the class of -induced minor-free graphs is well-quasi-ordered by the
induced minor relation if and only if is an induced minor of the gem (the
path on 4 vertices plus a dominating vertex) or of the graph obtained by adding
a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved
two decomposition theorems which are of independent interest.
Similar dichotomy results were previously given for subgraphs by Guoli Ding
in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502,
1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and
well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]
Simplicial decompositions of graphs: a survey of applications
AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects
Minor-Obstructions for Apex-Pseudoforests
A graph is called a pseudoforest if none of its connected components contains
more than one cycle. A graph is an apex-pseudoforest if it can become a
pseudoforest by removing one of its vertices. We identify 33 graphs that form
the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of
all minor-minimal graphs that are not apex-pseudoforests
Balanced-chromatic number and Hadwiger-like conjectures
Motivated by different characterizations of planar graphs and the 4-Color
Theorem, several structural results concerning graphs of high chromatic number
have been obtained. Toward strengthening some of these results, we consider the
\emph{balanced chromatic number}, , of a signed graph
. This is the minimum number of parts into which the vertices of a
signed graph can be partitioned so that none of the parts induces a negative
cycle. This extends the notion of the chromatic number of a graph since
, where denotes the signed graph
obtained from~ by replacing each edge with a pair of (parallel) positive and
negative edges. We introduce a signed version of Hadwiger's conjecture as
follows.
Conjecture: If a signed graph has no negative loop and no
-minor, then its balanced chromatic number is at most .
We prove that this conjecture is, in fact, equivalent to Hadwiger's
conjecture and show its relation to the Odd Hadwiger Conjecture.
Motivated by these results, we also consider the relation between
subdivisions and balanced chromatic number. We prove that if has
no negative loop and no -subdivision, then it admits a balanced
-coloring. This qualitatively generalizes a result of
Kawarabayashi (2013) on totally odd subdivisions
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