9 research outputs found
Double Roman domination and domatic numbers of graphs
A double Roman dominating function on a graph with vertex set is defined in \cite{bhh} as a function
having the property that if , then the vertex must have at least two
neighbors assigned 2 under or one neighbor with , and if , then the vertex must have
at least one neighbor with . The weight of a double Roman dominating function is the sum
, and the minimum weight of a double Roman dominating function on is the double Roman
domination number of .
A set of distinct double Roman dominating functions on with the property that
for each is called in \cite{v} a double Roman dominating family (of functions)
on . The maximum number of functions in a double Roman dominating family on is the double Roman domatic number
of .
In this note we continue the study of the double Roman domination and domatic numbers. In particular, we present
a sharp lower bound on , and we determine the double Roman domination and domatic numbers of some
classes of graphs
Upper bounds for covering total double Roman domination
Let G = (V, E) be a finite simple graph where V = V (G) and E = E(G). Suppose that G has no isolated vertex. A covering total double Roman dominating function (CT DRD function) f of G is a total double Roman dominating function (T DRD function) of G for which the set {v ∈ V (G)|f(v) ≠ 0} is a covering set. The covering total double Roman domination number γctdR(G) is the minimum weight of a CT DRD function on G. In this work, we present some contributions to the study of γctdR(G)-function of graphs. For the non star trees T, we show that γctdR(T) ≤ 4n(T )+5s(T )−4l(T )/3, where n(T), s(T) and l(T) are the order, the number of support vertices and the number of leaves of T respectively. Moreover, we characterize trees T achieve this bound. Then we study the upper bound of the 2-edge connected graphs and show that, for a 2-edge connected graphs G, γctdR(G) ≤ 4n/3 and finally, we show that, for a simple graph G of order n with δ(G) ≥ 2, γctdR(G) ≤ 4n/3 and this bound is sharp.Publisher's Versio
A new approach on locally checkable problems
By providing a new framework, we extend previous results on locally checkable
problems in bounded treewidth graphs. As a consequence, we show how to solve,
in polynomial time for bounded treewidth graphs, double Roman domination and
Grundy domination, among other problems for which no such algorithm was
previously known. Moreover, by proving that fixed powers of bounded degree and
bounded treewidth graphs are also bounded degree and bounded treewidth graphs,
we can enlarge the family of problems that can be solved in polynomial time for
these graph classes, including distance coloring problems and distance
domination problems (for bounded distances)
Advances in Discrete Applied Mathematics and Graph Theory
The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” 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
Some Progress on the Double Roman Domination in Graphs
For a graph G = (V,E), a double Roman dominating function (or just DRDF) is a function f : V → {0, 1, 2, 3} having the property that if f(v) = 0 for a vertex v, then v has at least two neighbors assigned 2 under f or one neighbor assigned 3 under f, and if f(v) = 1, then vertex v must have at least one neighbor ω with f(ω) ≥ 2. The weight of a DRDF f is the sum f(V ) = Σv∈Vf(v), and the minimum weight of a DRDF on G is the double Roman domination number of G, denoted by γdR(G). In this paper, we derive sharp upper and lower bounds on γdR(G) + γdR(Ḡ) and also γdR(G)γdR(Ḡ) ,where Ḡ is the complement of graph G. We also show that the decision problem for the double Roman domination number is NP- complete even when restricted to bipartite graphs and chordal graphs