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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

Daniël Paulusma, Christophe Picouleau, Bernard Ries
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]

Daniël Paulusma and Christophe Picouleau and Bernard Ries
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

Christophe Picouleau, Daniël Paulusma, Bernard Ries
2017 Electronic Notes in Discrete Mathematics
Abstract 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

Ling Xue, Jianliang Wu
2013 Open Journal of Discrete Mathematics
It is proved here that if 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

Boštjan Brešar, Sandi Klavžar
2007 Discrete Mathematics
This implies 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]

Dainis Zeps
2012 arXiv   pre-print
We build unbounded classes of plane and projective plane multiwheels that are 4-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]

Tao Wang and Ping Wang and Qinglin Yu
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

Eko Budi Santoso
2018 International Journal of Indonesian Education and Teaching
Some concepts in 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

Eko Santoso
2012 International Journal of Indonesian Education and Teaching
Some concepts in 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

Jing Huang, Anders Yeo
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

William F. Klostermeyer, Gary MacGillivray
2017 Discussiones Mathematicae Graph Theory
The study of the stronger assertion that such 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

Claude Berge
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  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]

José Antonio Martín H
2011 arXiv   pre-print
In this paper I study 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]

Hubert de Fraysseix, Patrice Ossona de Mendez
2002 Lecture Notes in Computer Science
We give 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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