The domatic numbers of a graph $G$ and of its complement $\bar{G}$ were studied by J. E. Dunbar, T. W. Haynes and M. A. Henning. They suggested four open problems. We will solve the following ones: Characterize bipartite graphs $G$ having $d(G) = d(\bar{G})$. Further, we will present a partial solution to the problem: Is it true that if $G$ is a graph satisfying $d(G) = d(\bar{G})$, then $\gamma (G) = \gamma (\bar{G})$? Finally, we prove an existence theorem concerning the total domatic number of a graph and of its complement.
Generalized Petersen graphs are certain graphs consisting of one quadratic factor. For these graphs some numerical invariants concerning the domination are studied, namely the domatic number $d(G)$, the total domatic number $d_t(G)$ and the $k$-ply domatic number $d^k(G)$ for $k=2$ and $k=3$. Some exact values and some inequalities are stated.
For any graph G, let V (G) and E(G) denote the vertex set and the edge set of G respectively. The Boolean function graph B(G, L(G), NINC) of G is a graph with vertex set V (G) ∪ E(G) and two vertices in B(G, L(G), NINC) are adjacent if and only if they correspond to two adjacent vertices of G, two adjacent edges of G or to a vertex and an edge not incident to it in G. For brevity, this graph is denoted by B1(G). In this paper, we determine domination number, independent, connected, total, cycle, point-set, restrained, split and non-split domination numbers of B1(G) and obtain bounds for the above numbers.
For any graph G, let V (G) and E(G) denote the vertex set and the edge set of G respectively. The Boolean function graph B(G, L(G), NINC) of G is a graph with vertex set V (G) ∪ E(G) and two vertices in B(G, L(G), NINC) are adjacent if and only if they correspond to two adjacent vertices of G, two adjacent edges of G or to a vertex and an edge not incident to it in G. For brevity, this graph is denoted by B1(G). In this paper, we determine domination number, independent, connected, total, point-set, restrained, split and non-split domination numbers in the complement B1(G) of B1(G) and obtain bounds for the above numbers.
Microbial mats in hot springs form a dynamic ecosystem and support the growth of diverse communities with broad-ranging metabolic capacity. In this study, we used 16S rRNA gene amplicon sequencing to analyse microbial communities in mat samples from two hot springs in Al Aridhah, Saudi Arabia. Putative metabolic pathways of the microbial communities were identified using phylogenetic investigation of communities by reconstruction of unobserved states (PICRUSt). Filamentous anoxygenic phototrophic bacteria associated with phylum Chloroflexi were abundant (> 50 %) in both hot springs at 48 °C. Chloroflexi were mainly represented by taxa Chloroflexus followed by Roseiflexus. Cyanobacteria of genus Arthrospira constituted 3.4 % of microbial mats. Heterotrophic microorganisms were mainly represented by Proteobacteria, Actinobacteria, Bacteroidetes, and Firmicutes. Archaea were detected at a lower relative abundance (< 1 %). Metabolic pathways associated with membrane transport, carbon fixation, methane metabolism, amino acid biosynthesis, and degradation of aromatic compounds were commonly found in microbial mats of both hot springs. In addition, pathways for production of secondary metabolites and antimicrobial compounds were predicted to be present in microbial mats. In conclusion, microbial communities in the hot springs of Al Aridhah were composed of diverse bacteria, with taxa of Chloroflexus being dominant.
For a graphical property P and a graph G, a subset S of vertices of G is a P-set if the subgraph induced by S has the property P. The domination number with respect to the property P, is the minimum cardinality of a dominating P-set. In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate and hereditary properties when a graph is modified by adding an edge or deleting a vertex.
In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate property P, denoted by γP (G), when a graph G is modified by deleting a vertex or deleting edges. A graph G is (γP (G), k)P -critical if γP (G − S) < γP (G) for any set S ( V (G) with |S| = k. Properties of (γP , k)P -critical graphs are studied. The plus bondage number with respect to the property P, denoted b + P (G), is the cardinality of the smallest set of edges U ⊆ E(G) such that γP (G − U) > γP (G). Some known results for ordinary domination and bondage numbers are extended to γP (G) and b + P (G). Conjectures concerning b + P (G) are posed.