10 research outputs found
A Greedy Partition Lemma for Directed Domination
A directed dominating set in a directed graph is a set of vertices of
such that every vertex has an adjacent vertex
in with directed to . The directed domination number of , denoted
by , is the minimum cardinality of a directed dominating set in .
The directed domination number of a graph , denoted , which is
the maximum directed domination number over all orientations of
. The directed domination number of a complete graph was first studied by
Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this
paper we prove a Greedy Partition Lemma for directed domination in oriented
graphs. Applying this lemma, we obtain bounds on the directed domination
number. In particular, if denotes the independence number of a graph
, we show that .Comment: 12 page
On the structure of graphs with forbidden induced substructures
One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints.
In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs.
ErdĆs and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every -edge-colouring of the complete graph on vertices there is a monochromatic clique on at least vertices. The famous ErdĆs-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs.
In the second part of this thesis we focus more on order-size pairs; an order-size pair is the family consisting of all graphs of order and size , i.e. on vertices with edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs , i.e. for approaching infinity, the limit superior of the fraction of all possible sizes , such that the order-size pair does not avoid the pair
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
Advances in Discrete Applied Mathematics and Graph Theory
The present reprint contains twelve papers published in the Special Issue âAdvances in Discrete Applied Mathematics and Graph Theory, 2021â of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs