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A characterization of locating-total domination edge critical graphs

2011
*
Discussiones Mathematicae Graph Theory
*

*A*

*graph*G is said to be

*a*locating-total domination

*edge*removal

*critical*

*graph*, or just

*a*γ t+ L -ER-

*critical*

*graph*, if γ t L (G − e) > γ t L (G) for all e non-pendant

*edge*of E. ... For

*a*

*graph*G = (V, E)

*without*isolated vertices,

*a*subset D of vertices of V is

*a*total dominating set (TDS) of G if every

*vertex*in V is adjacent to

*a*

*vertex*in D. ...

*A*

*graph*G is said to be

*a*locating-total domination

*edge*removal

*critical*

*graph*, or just

*a*γ t+ L -ER-

*critical*

*graph*, if γ t L (G − e) > γ t L (G) for all e non- pendant

*edge*of E. ...

##
###
Critical vertices and edges in H-free graphs

2019
*
Discrete Applied Mathematics
*

*A*

*vertex*or

*edge*in

*a*

*graph*is

*critical*if its deletion reduces the chromatic number of the

*graph*by one. ... We consider the problems of deciding whether

*a*

*graph*has

*a*

*critical*

*vertex*or

*edge*, respectively. ... In this paper we consider the problems

*Critical*

*Vertex*and

*Critical*

*Edge*, which are to decide if

*a*

*graph*has

*a*

*critical*

*vertex*or

*critical*

*edge*, respectively. ...

##
###
Critical Vertices and Edges in H-free Graphs
[article]

2017
*
arXiv
*
pre-print

*A*

*vertex*or

*edge*in

*a*

*graph*is

*critical*if its deletion reduces the chromatic number of the

*graph*by 1. ... We consider the problems of deciding whether

*a*

*graph*has

*a*

*critical*

*vertex*or

*edge*, respectively. ...

*A*

*graph*is

*vertex*-

*critical*if every

*vertex*is

*critical*and

*edge*-

*critical*if every

*edge*is

*critical*. ...

##
###
Reducing the Chromatic Number by Vertex or Edge Deletions

2017
*
Electronic Notes in Discrete Mathematics
*

Abstract

doi:10.1016/j.endm.2017.10.042
fatcat:l6j22gchyveotcp27okbpisjbe
*A**vertex*or*edge*in*a**graph*is*critical*if its deletion reduces the chromatic number of the*graph*by 1. ... We consider the problems of testing whether*a**graph*has*a**critical**vertex*or*edge*, respectively. ... We call the problems of deciding if*a**graph*has*a**critical**vertex*,*critical**edge*or contraction-*critical**edge**Critical**Vertex*,*Critical**Edge*and Contraction-*Critical**Edge*, respectively. ...##
###
Edge Colorings of Planar Graphs without 6-Cycles with Two Chords

2013
*
Open Journal of Discrete Mathematics
*

It is proved here that if

doi:10.4236/ojdm.2013.32016
fatcat:d3e7sjhxsrezthidsezunitlau
*a*planar*graph*has maximum degree at least 6 and any 6-cycle contains at most one chord, then it is of class 1. ...*A**critical**graph*with maximum degree is called*a*-*critical**graph*. It is clear that every*critical**graph*is 2-connected. ... Introduction All*graphs*considered here are finite and simple. Let G be*a**graph*with the*vertex*set and*edge*set . ...##
###
Maximal proper subgraphs of median graphs

2007
*
Discrete Mathematics
*

This implies

doi:10.1016/j.disc.2005.11.076
fatcat:z2esvqf7pvcrdeeqi5avfff6yq
*a*characterization of*vertex*-*critical*(respectively,*vertex*-complete) median*graphs*, which are median*graphs*whose all*vertex*-deleted subgraphs are not median (respectively, are median). ... For*a*median*graph*G and*a**vertex*v of G that is not*a*cut-*vertex*we show that G − v is*a*median*graph*precisely when v is not the center of*a*bipartite wheel, which is in turn equivalent with the existence ... Fig. 1 . 1*A**vertex*-*critical*but not*edge*-*critical*median*graph*. ...##
###
On building 4-critical plane and projective plane multiwheels from odd wheels
[article]

2012
*
arXiv
*
pre-print

We build unbounded classes of plane and projective plane multiwheels that are 4-

arXiv:1202.4862v1
fatcat:24iipbchb5dkxnx62vd4lj4ux4
*critical*that are received summing odd wheels as*edge*sums modulo two. ... These classes can be considered as ascending from single common*graph*that can be received as*edge*sum modulo two of the octahedron*graph*O and the minimal wheel W3. ... Lemma 4 Let by summation of*edge*sets of wheels some wheels intersect in vertices*without*incident*edges*. Then resulting*graph*can't be 4-*critical*. ...##
###
On Murty-Simon Conjecture II
[article]

2013
*
arXiv
*
pre-print

*A*

*graph*is diameter two

*edge*-

*critical*if its diameter is two and the deletion of any

*edge*increases the diameter. ... Murty and Simon conjectured that the number of

*edges*in

*a*diameter two

*edge*-

*critical*

*graph*on n vertices is at most n^2/4 and the extremal

*graph*is the complete bipartite

*graph*K_n/2, n/2. ... If G is

*a*3-γ t -

*edge*

*critical*

*graph*, then 2 ≤ diam(G) ≤ 3. ...

##
###
MATHEMATICS CLASSROOM ACTIVITIES BASED ON SOME TOPICS IN GRAPH THEORY TO DEVELOP CRITICAL THINKING OF PRIMARY AND SECONDARY SCHOOL STUDENTS

2018
*
International Journal of Indonesian Education and Teaching
*

Some concepts in

doi:10.24071/ijiet.v2i2.921
fatcat:52euzmrai5ccjaso62pvd65vlm
*graph*theory, such as*vertex*coloring and domination set can be used in primary and secondary schools to develop*critical*thinking of the students. ... This paper is*a*theoretical study in Mathematics Education which proposes some Mathematics classroom activities based on some concepts in*graph*theory to develop*critical*thinking for high school students ... Given*a**graph*, we ask the student to draw*a*similar*graph**without*taking*a*pen from the paper and*without*retracing the same*edge*, begin and end at the same*vertex*. ...##
###
MATHEMATICS CLASSROOM ACTIVITIES BASED ON SOME TOPICS IN GRAPH THEORY TO DEVELOP CRITICAL THINKING OF PRIMARY AND SECONDARY SCHOOL STUDENTS

2012
*
International Journal of Indonesian Education and Teaching
*

Some concepts in

doi:10.24071/ijiet.2018.020207
fatcat:po36nw2r4rg7jbqft4txae3sgi
*graph*theory, such as*vertex*coloring and domination set can be used in primary and secondary schools to develop*critical*thinking of the students. ... This paper is*a*theoretical study in Mathematics Education which proposes some Mathematics classroom activities based on some concepts in*graph*theory to develop*critical*thinking for high school students ... Given*a**graph*, we ask the student to draw*a*similar*graph**without*taking*a*pen from the paper and*without*retracing the same*edge*, begin and end at the same*vertex*. ...##
###
Maximal and Minimal Vertex-Critical Graphs of Diameter Two

1998
*
Journal of combinatorial theory. Series B (Print)
*

*A*

*graph*is

*vertex*-

*critical*(

*edge*-

*critical*) if deleting any

*vertex*(

*edge*) increases its diameter. ... Royle for providing the

*graph*in Figure 5 (

*a*) and Dr. L. Caccetta for his interest and some interesting discussions. ... The Petersen

*graph*is

*a*2-

*vertex*-

*critical*

*graph*with 10 vertices and 15 edges.Figure 5(

*a*) shows

*a*2-

*vertex*-

*critical*

*graph*with 12 vertices and 21

*edges*andFigure 5(b) shows how to construct

*a*2-

*vertex*-

*critical*...

##
###
Eternal domination: criticality and reachability

2017
*
Discussiones Mathematicae Graph Theory
*

The study of the stronger assertion that such

doi:10.7151/dmgt.1918
fatcat:veeynfzrfrcovh76nqr6264riy
*a*set can be reached after*a*single attack is defended leads to the study of*graphs*which are*critical*in the sense that deleting any*vertex*reduces the eternal ... We show that for every minimum eternal dominating set, D, of*a**graph*G and every*vertex*v ∈ D, there is*a*sequence of attacks at the vertices of G which can be defended in such*a*way that an eternal dominating ... Proposition 3.10 If G is*a*connected eternal domination*vertex*-*critical**graph*with at least one*edge*, then κ (G) ≥ 2. Proof. ...##
###
Some properties of non-bicolorable hypergraphs and the four-color problem

1996
*
Discrete Applied Mathematics
*

*A*non-bicolorable hypergraph which becomes bicolorable when any of its

*edges*is removed is called "

*edge*-

*critical*". and several of its propertics can be found in the literature (C. ... In this paper, instead of

*edge*-

*critical*hypergraphs. we study the

*vertex*-

*critical*hypergraphs; the applications are more numerous. and it appears that somewhat stronger results could imply the famous " ... Seymour [14] has characterized the

*edge*-

*critical*hypergraphs having as many vertices as

*edges*(by association with strongly connected directed

*graphs*

*without*even circuits). ...

##
###
Minimal non-extensible precolorings and implicit-relations
[article]

2011
*
arXiv
*
pre-print

In this paper I study

arXiv:1104.0510v1
fatcat:pgrkknib45cd7bukhjd2z65twq
*a*variant of the general*vertex*coloring problem called precoloring. ... It is interesting per se that, for*graphs*of arbitrarily large chromatic number, the minimal number of colored vertices, in*a*non-extensible precoloring, remains constant; only two vertices u,v suffice ... If all the vertices of*a**graph*G are*critical*we say that G is*vertex*-*critical*and if every element (*vertex*or*edge*) of G is*critical*we say that G is*a**critical**graph*and more specifically if χ(G) = k ...##
###
A Characterization of DFS Cotree Critical Graphs
[chapter]

2002
*
Lecture Notes in Computer Science
*

We give

doi:10.1007/3-540-45848-4_7
fatcat:ppok5v4xunfhpl3uwj4cmhhqfa
*a*characterization of DFS-cotree*critical**graphs*which is central to the linear time Kuratowski finding algorithm implemented in PIGALE 1 . ... Every cotree*critical**graph*is 3-connected. Proof. Let G be*a*cotree*critical**graph*. Assume G has*a*cut-*vertex*v. ... Similarly, subdividing an*edge*creates two*edges*with the same status*without*changing the status of the other*edges*and hence cannot create*a*cycle of non*critical**edges*. Definition 4. ...
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