11 research outputs found
Changing upper irredundance by edge addition
AbstractDenote the upper irredundance number of a graph G by IR(G). A graph G is IR-edge-addition-sensitive if its upper irredundance number changes whenever an edge of Ḡ is added to G. Specifically, G is IR-edge-critical (IR+-edge-critical, respectively) if IR(G+e)<IR(G) (IR(G+e)>IR(G), respectively) for each edge e of Ḡ. We show that if G is IR-edge-addition-sensitive, then G is either IR-edge-critical or IR+-edge-critical. We obtain properties of the latter class of graphs, particularly in the case where β(G)=IR(G)=2 (where β(G) denotes the vertex independence number of G). This leads to an infinite class of IR+-edge-critical graphs where IR(G)=2
Critical concepts in domination, independence and irredundance of graphs
The lower and upper independent, domination and irredundant numbers of the graph
G = (V, E) are denoted by i ( G) , f3 ( G), 'Y ( G), r ( G), ir ( G) and IR ( G) respectively.
These six numbers are called the domination parameters. For each of these parameters
n:, we define six types of criticality. The graph G is n:-critical (n:+ -critical) if the
removal of any vertex of G causes n: (G) to decrease (increase), G is n:-edge-critical
(n:+-edge-critical) if the addition of any missing edge causes n: (G) to decrease (increase),
and G is Ir-ER-critical (n:- -ER-critical) if the removal of any edge causes
n: (G) to increase (decrease). For all the above-mentioned parameters n: there exist
graphs which are n:-critical, n:-edge-critical and n:-ER-critical. However, there do not
exist any n:+-critical graphs for n: E {ir,"f,i,/3,IR}, no n:+-edge-critical graphs for
n: E {ir,"f,i,/3} and non:--ER-critical graphs for: E {'Y,/3,r,IR}. Graphs which
are "I-critical, i-critical, "I-edge-critical and i-edge-critical are well studied in the literature.
In this thesis we explore the remaining types of criticality.
We commence with the determination of the domination parameters of some wellknown
classes of graphs. Each class of graphs we consider will turn out to contain a
subclass consisting of graphs that are critical according to one or more of the definitions
above. We present characterisations of "I-critical, i-critical, "I-edge-critical and
i-edge-critical graphs, as well as ofn:-ER-critical graphs for n: E {/3,r,IR}. These
characterisations are useful in deciding which graphs in a specific class are critical.
Our main results concern n:-critical and n:-edge-critical graphs for n: E {/3, r, IR}. We show that the only /3-critical graphs are the edgeless graphs and that a graph is IRcritical
if and only if it is r-critical, and proceed to investigate the r-critical graphs
which are not /3-critical. We characterise /3-edge-critical and r-edge-critical graphs
and show that the classes of IR-edge-critical and r-edge-critical graphs coincide. We also exhibit classes of r+ -critical, r+ -edge-critical and i- -ER-critical graphs.Mathematical SciencesD. Phil. (Mathematics
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Aspects of distance and domination in graphs.
Thesis (Ph.D.-Mathematics and Applied Mathematics)-University of Natal, 1995.The first half of this thesis deals with an aspect of domination; more specifically, we
investigate the vertex integrity of n-distance-domination in a graph, i.e., the extent
to which n-distance-domination properties of a graph are preserved by the deletion
of vertices, as well as the following: Let G be a connected graph of order p and let
oi- S s;:; V(G). An S-n-distance-dominating set in G is a set D s;:; V(G) such that
each vertex in S is n-distance-dominated by a vertex in D. The size of a smallest
S-n-dominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G),
then S is called an n-distance-domination-forcing set of G, and the cardinality of a
smallest n-distance-domination-forcing set of G is denoted by On(G). We investigate
the value of On(G) for various graphs G, and we characterize graphs G for which
On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value,
namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept
of n-distance-domination of vertices (above) by the concept of the covering of edges
is also investigated.
For k E {a, 1, ... ,rad(G)}, the set S is said to be a k-radius-forcing set if, for each
v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest
k-radius-forcing set of G is called the k-radius-forcing number of G and is denoted
by Pk(G). We investigate the value of Prad(G) for various classes of graphs G,
and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values.
We show that the problem of determining Pk(G) is NP-complete, study the
sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship
between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity
of G, for an edge e of the complement of G.
Finally, we characterize integral triples representing realizable values of the triples
b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph
Two conjectures on 3-domination critical graphs
For a graph G = (V (G), E (G)), a set S ~ V (G) dominates G if each vertex
in V (G) \S is adjacent to a vertex in S. The domination number I (G) (independent
domination number i (G)) of G is the minimum cardinality amongst its dominating
sets (independent dominating sets). G is k-edge-domination-critical, abbreviated k-1-
critical, if the domination number k decreases whenever an edge is added. Further, G
is hamiltonian if it has a cycle that passes through each of its vertices.
This dissertation assimilates research generated by two conjectures:
Conjecture I. Every 3-1-critical graph with minimum degree at least two is hamiltonian.
Conjecture 2. If G is k-1-critical, then I ( G) = i ( G).
The recent proof of Conjecture I is consolidated and presented accessibly. Conjecture
2 remains open for k = 3 and has been disproved for k :::>: 4. The progress is
detailed and proofs of new results are presented.Mathematical ScienceM. Sc. (Mathematics
Distances and Domination in Graphs
This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present