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40
A dynamic survey of graph labellings
 Electron. J. Combin., Dynamic Surveys(6):95pp
, 2001
"... A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done ..."
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Cited by 168 (0 self)
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A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.
Applications and Variations of Domination in Graphs
, 2000
"... In a graph G =(V,E), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively. This dissertation explores some new variations and applications of dominating sets. We first introduce the ..."
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Cited by 19 (0 self)
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In a graph G =(V,E), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively. This dissertation explores some new variations and applications of dominating sets. We first introduce the concept of Roman domination. A Roman dominating function is a function f: V →{0, 1, 2} such that every vertex v for which f(v) =0hasa neighbor w with f(w) = 2. This corresponds to a problem in army placement where every region is either defended by its own army or has a neighbor with two armies, in which case one of the two armies can be sent to the undefended region if a conflict breaks out. The weight of a Roman dominating function f is f(V) = � v∈V f(v), and we are interested in finding Roman dominating functions of minimum weight. We explore the graph theoretic, algorithmic, and complexity issues of Roman domination, including algorithms for finding minimum weight Roman dominating functions for trees and grids.
CONDITIONAL RESOLVABILITY IN GRAPHS: A SURVEY
, 2004
"... For an ordered setW = {w1,w2,...,wk} of vertices and a vertexv in a connected graphG, the code of v with respect toW is the kvector cW (v) = (d(v,w1),d(v,w2),...,d(v,wk)), where d(x,y) represents the distance between the vertices x andy. The setW is a resolving set for G if distinct vertices of G h ..."
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Cited by 17 (0 self)
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For an ordered setW = {w1,w2,...,wk} of vertices and a vertexv in a connected graphG, the code of v with respect toW is the kvector cW (v) = (d(v,w1),d(v,w2),...,d(v,wk)), where d(x,y) represents the distance between the vertices x andy. The setW is a resolving set for G if distinct vertices of G have distinct codes with respect to W. The minimum cardinality of a resolving set for G is its dimension dim(G). Many resolving parameters are formed by extending resolving sets to different subjects in graph theory, such as the partition of the vertex set, decomposition, and coloring in graphs, or by combining resolving property with another graphtheoretic property such as being connected, independent, or acyclic. In this paper, we survey results and open questions on the resolving parameters defined by imposing an additional constraint on resolving sets, resolving partitions, or resolving decompositions in graphs.
The Complexity of GFree Colourability
 Discrete Math
, 1997
"... The problem of determining if a graph is 2colourable (i.e., bipartite) has long been known to have a simple polynomial time algorithm. Being 2colourable is equivalent to having a bipartition of the vertex set where each cell is K 2 free. We extend this notion to determining if there exists a bipa ..."
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Cited by 10 (1 self)
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The problem of determining if a graph is 2colourable (i.e., bipartite) has long been known to have a simple polynomial time algorithm. Being 2colourable is equivalent to having a bipartition of the vertex set where each cell is K 2 free. We extend this notion to determining if there exists a bipartition where each cell is Gfree for some fixed graph G. One might expect that for some graphs other than K 2 ; K 2 there also exist polynomial time algorithms. Rather surprisingly we show that for any graph G on more than two vertices the problem is NPcomplete. 1 Introduction A vertex kcolouring of a graph is an assignment of one of k colours to each vertex such that adjacent vertices receive different colours. Such colourings have been studied extensively and form one of the oldest and deepest areas of graph theory. In this course of study many generalisations of the colouring concept have been suggested. The following two notions, introduced in [Har85], appear to be useful in expres...
On the Minors of an Incidence Matrix and its Smith Normal Form
"... Consider the vertexedge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of vertexdisjoint odd cycles (i.e., the odd tulgeity) of the graph. The problem of determining this number is shown ..."
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Cited by 9 (1 self)
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Consider the vertexedge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of vertexdisjoint odd cycles (i.e., the odd tulgeity) of the graph. The problem of determining this number is shown to be NPhard. Turning to maximal minors, we determine the rank of the incidence matrix. This depends on the number of components of the graph containing no odd cycle. We then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix, as well as its Smith normal form. These results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs. Finally, we give a sufficient condition for a system of equations (whose coefficient matrix is an incidence matrix) to admit an integer solution.
Some structural, metric and convex properties of the boundary of a graph
"... Let u, v be two vertices of a connected graph G. The vertex v is said to be a boundary vertex of u if no neighbor of v is further away from u than v. The boundary of a graph is the set of all its boundary vertices. In this work, we present a number of properties of the boundary of a graph under diff ..."
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Cited by 7 (0 self)
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Let u, v be two vertices of a connected graph G. The vertex v is said to be a boundary vertex of u if no neighbor of v is further away from u than v. The boundary of a graph is the set of all its boundary vertices. In this work, we present a number of properties of the boundary of a graph under different points of view: (1) a realization theorem involving different types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary; (2) the contour is a monophonic set; and (3) the cardinality of the boundary is an upper bound for both the metric dimension and the determining number of a graph.
Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and paths make things difficult
, 2003
"... We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture... ..."
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Cited by 6 (0 self)
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We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture...
Iterations of Eccentric Digraphs
 BULL. INST. COMBIN. APPL
"... The eccentricity e(u) of vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a digraph G is the digraph that has the same vertex set as G and the arc set defined ..."
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Cited by 5 (1 self)
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The eccentricity e(u) of vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a digraph G is the digraph that has the same vertex set as G and the arc set defined by: there is an arc from u to v if and only if v is an eccentric vertex of u. In this paper we consider the behaviour of an iterated sequence of eccentric graphs or digraphs of a graph or a digraph. The paper concludes with several open problems.
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 4 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
Conditional Coloring
, 1998
"... The conditional chromatic number Ø (G; P ) of a graph G with respect to a graphical property P is the minimum number of colors needed to color the vertices of G such that each color class induces a subgraph of G with property P . When P is the property that a graph contains no subgraph isomorphic t ..."
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Cited by 3 (0 self)
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The conditional chromatic number Ø (G; P ) of a graph G with respect to a graphical property P is the minimum number of colors needed to color the vertices of G such that each color class induces a subgraph of G with property P . When P is the property that a graph contains no subgraph isomorphic to a graph F , we write Ø (G; :F ). The conditional chromatic number of a graph has been studied by various authors since 1968. We focus on two conditional chromatic numbers, specifically Ø (G; :C j ) and Ø (G; :P j ), where C j is a cycle of length j for some fixed j 3 and P j is a path of length j \Gamma 1 for some fixed j 2. We find Ø (G; :C j ) for graphs missing at most j \Gamma 1 edges and Ø (G; :P j ) for graphs missing at most 2j \Gamma 5 edges. To accomplish this, we characterize all Hamiltonian graphs of order n with at least i n 2 j \Gamma (n \Gamma 1) edges and all graphs with no Hamiltonian paths with at least i n 2 j \Gamma (2n \Gamma 5) edges. We determine...