70 research outputs found
On the integrity of domination in graphs.
Thesis (M.Sc.)-University of Natal, 1993.This thesis deals with an investigation of the integrity of domination in a.graph, i.e., the extent to
which domination properties of a graph are preserved if the graph is altered by the deletion of
vertices or edges or by the insertion of new edges.
A brief historical introduction and motivation are provided in Chapter 1. Chapter 2 deals with kedge-(
domination-)critical graphs, i.e., graphsG such that )'(G) = k and )'(G+e) < k for all e E
E(G). We explore fundamental properties of such graphs and their characterization for small
values of k. Particular attention is devoted to 3-edge-critical graphs.
In Chapter 3, the changes in domination number brought aboutby vertex removal are investigated.
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Parameters )'+'(G) (and "((G)), denoting the smallest number of vertices of G in a set 5 such that
)'(G-5) > )'(G) ()'(G -5) < )'(G), respectively), are investigated, as are'k-vertex-critical graphs G
(with )'(G) = k and )'(G-v) < k for all v E V(O)). The existence of smallest'domination-forcing
sets of vertices of graphs is considered.
The bondage number 'Y+'(G), i.e., the smallest number of edges of a graph G in a set F such that
)'(G- F) > )'(0), is investigated in Chapter 4, as are associated extremal graphs. Graphs with
dominating sets or domination numbers that are insensitive to the removal of an arbitrary edge are
considered, with particular reference to such graphs of minimum size.
Finally, in Chapter 5, we-discuss n-dominating setsD of a graph G (such that each vertex in G-D
is adjacent to at least n vertices in D) and associated parameters. All chapters but the first and
fourth contain a listing of unsolved problems and conjectures
Cyclic edge extensions-self centered graphs
The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. The maximum
and the minimum eccentricity among the vertices of a graph G are known as the diameter and the radius
of G respectively. If they are equal then the graph is said to be a self - centered graph. Edge addition
/extension to a graph either retains or changes the parameter of a graph, under consideration. In this
paper mainly, we consider edge extension for cycles, with respect to the self-centeredness(of cycles),that
is, after an edge set is added to a self centered graph the resultant graph is also a self-centered graph.
Also, we have other structural results for graphs with edge -extensions
Complexity of Stability
Graph parameters such as the clique number, the chromatic number, and the
independence number are central in many areas, ranging from computer networks
to linguistics to computational neuroscience to social networks. In particular,
the chromatic number of a graph (i.e., the smallest number of colors needed to
color all vertices such that no two adjacent vertices are of the same color)
can be applied in solving practical tasks as diverse as pattern matching,
scheduling jobs to machines, allocating registers in compiler optimization, and
even solving Sudoku puzzles. Typically, however, the underlying graphs are
subject to (often minor) changes. To make these applications of graph
parameters robust, it is important to know which graphs are stable for them in
the sense that adding or deleting single edges or vertices does not change
them. We initiate the study of stability of graphs for such parameters in terms
of their computational complexity. We show that, for various central graph
parameters, the problem of determining whether or not a given graph is stable
is complete for \Theta_2^p, a well-known complexity class in the second level
of the polynomial hierarchy, which is also known as "parallel access to NP.
Recognizing Maximal Unfrozen Graphs with respect to Independent Sets is CO-NP-complete
A graph is unfrozen with respect to k independent set if it has an independent set of size k after the addition of any edge. The problem of recognizing such graphs is known to be NP-complete. A graph is maximal if the addition of one edge means it is no longer unfrozen. We designate the problem of recognizing maximal unfrozen graphs as MAX(U(k-SET)) and show that this problem is CO-NP-complete. This partially fills a gap in known complexity cases of maximal NP-complete problems, and raises some interesting open conjectures discussed in the conclusion
Techniques pour l'exploration de données structurées et pour la découverte de connaissances en théorie des graphes
Improving frequent subgraph mining in the presence of symmetry -- Using background knowledge to improve structured data mining -- Automated generation of conjectures on forbidden subgraph characterization
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..
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