118 research outputs found
Orientable domination in product-like graphs
The orientable domination number, , of a graph is the
largest domination number over all orientations of . In this paper, is studied on different product graphs and related graph operations. The
orientable domination number of arbitrary corona products is determined, while
sharp lower and upper bounds are proved for Cartesian and lexicographic
products. A result of Chartrand et al. from 1996 is extended by establishing
the values of for arbitrary positive integers
and . While considering the orientable domination number of
lexicographic product graphs, we answer in the negative a question concerning
domination and packing numbers in acyclic digraphs posed in [Domination in
digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022)
359-377]
Connectivity and other invariants of generalized products of graphs
Figueroa-Centeno et al. [4] introduced the following product of digraphs let D be a digraph and let G be a family of digraphs such that V (F) = V for every F¿G. Consider any function h:E(D)¿G. Then the product D¿hG is the digraph with vertex set V(D)×V and ((a,x),(b,y))¿E(D¿hG) if and only if (a,b)¿E(D) and (x,y)¿E(h(a,b)). In this paper, we deal with the undirected version of the ¿h-product, which is a generalization of the classical direct product of graphs and, motivated by the ¿h-product, we also recover a generalization of the classical lexicographic product of graphs, namely the °h-product, that was introduced by Sabidussi in 1961. We provide two characterizations for the connectivity of G¿hG that generalize the existing one for the direct product. For G°hG, we provide exact formulas for the connectivity and the edge-connectivity, under the assumption that V (F) = V , for all F¿G. We also introduce some miscellaneous results about other invariants in terms of the factors of both, the ¿h-product and the °h-product. Some of them are easily obtained from the corresponding product of two graphs, but many others generalize the existing ones for the direct and the lexicographic product, respectively. We end up the paper by presenting some structural properties. An interesting result in this direction is a characterization for the existence of a nontrivial decomposition of a given graph G in terms of ¿h-product.Postprint (author's final draft
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
On (2-d)-kernels in the cartesian product of graphs
In this paper we study the problem of the existence of (2-d)-kernels in the cartesian product of graphs. We give sufficient conditions for the existence of (2-d)-kernels in the cartesian product and also we consider the number of (2-d)-kernels
On the Packing Partitioning Problem on Directed Graphs
This work is aimed to continue studying the packing sets of digraphs via the perspective of
partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex
coloring of digraphs) and focused on finding the minimum cardinality among all packing partitions
for a given digraph D, called the packing partition number of D. Some lower and upper bounds on
this parameter are proven, and their exact values for directed trees are given in this paper. In the case
of directed trees, the proof results in a polynomial-time algorithm for finding a packing partition of
minimum cardinality. We also consider this parameter in digraph products. In particular, a complete
solution to this case is presented when dealing with the rooted products
Injective coloring of product graphs
The problem of injective coloring in graphs can be revisited through two
different approaches: coloring the two-step graphs and vertex partitioning of
graphs into open packing sets, each of which is equivalent to the injective
coloring problem itself. Taking these facts into account, we observe that the
injective coloring lies between graph coloring and domination theory.
We make use of these three points of view in this paper so as to investigate
the injective coloring of some well-known graph products. We bound the
injective chromatic number of direct and lexicographic product graphs from
below and above. In particular, we completely determine this parameter for the
direct product of two cycles. We also give a closed formula for the corona
product of two graphs
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