An independent Roman {2}-dominating function (IR2DF) f : V -> {0,1, 2} in a graph G = (V, E) has the properties that Sigma(u is an element of N(v)) f(u) >= 2 if f(v) = 0, and f(u) = 0 for u is an element of N(v) if f(v) >= 1, where v is an element of V. The weight of an IR2DF in a graph G is defined as the sum of its function value for all vertices, given by omega(f) = Sigma(v is an element of V) f(v). The independent Roman {2}-domination number of G, denoted i{R2}(G), is the minimum weight of an IR2DF on G. In this paper, we prove that the independent Roman {2}-domination problem (IR2D) is NP-complete, even when restricted to chordal bipartite graphs. We then give an exact formula for the IR2D in corona graphs. Finally, we present two linear-time algorithms for solving IR2D for proper interval graphs and block-cactus graphs, respectively.
Let G be a simple connected graph with diameter d, and k is an element of [1, d] be an integer. A radio k-coloring of graph G is a mapping g : V(G) -> {0} boolean OR N satisfying vertical bar g(u) - g(v)vertical bar >= 1 + k d(u, v) for any pair of distinct vertices u and v of the graph G, where d(u, v) denotes distance between vertices u and v in G. The number max{g(u) : u is an element of V (G)g is known as the span of g and is denoted by rc(k)(g). The radio k-chromatic number of graph G, denoted by rc(k)(G), is defined as min{rc(k)(g) : g is a radio k-coloring of G}. For k = d - 1 the radio k-coloring of graph G is called an antipodal coloring. So rc(d-1) is called the antipodal number of G and is denoted by ac(G). Here, we study antipodal coloring of the Cartesian product of the complete graph K-r and cycle C-s, K-r square C-s, for r >= 4 and s >= 3. We determine the antipodal number of K-r square C-s, for even r >= 4 with s 1 (mod 4), and for any r >= 4 with s = 4t + 2, t odd. Also, for the remaining values of r and s, we give lower and upper bounds for ac(K-r square C-s).
A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set {1, 2, ... , n} to the vertices of G. The strength str (G) of G is defined by str (G) = min {strf (G) |f is a numbering of G }, where strf (G) = max {f (u) + f (v) |uv is an element of E (G) }. Using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. To achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles.
A subset S of vertices of a graph G is in general position if no shortest path in G contains three vertices of S. The general position problem consists of finding the number of vertices in a largest general position set of G, whilst the lower general position problem asks for a smallest maximal general position set. In this paper we determine the lower general position numbers of several families of Cartesian products. We also show that the existence of small maximal general position sets in a Cartesian product is connected to a special type of general position set in the factors, which we call a terminal set, for which adding any vertex u from outside the set creates three vertices in a line with u as an endpoint. We give a constructive proof of the existence of terminal sets for graphs with diameter at most three. We also present conjectures on the existence of terminal sets for all graphs and a lower bound on the lower general position number of a Cartesian product in terms of the lower general position numbers of its factors.
For n >= 2t + 1 where t >= 1, the circulant graph C-n(1, 2, ... , t) consists of the vertices v(0), v(1), v(2), ... , v(n-1) and the edges v(i)v(i+1), v(i)v(i+2), ... , v(i)v(i+t), where i = 0, 1, 2, ... , n - 1, and the subscripts are taken modulo n. We prove that the metric dimension dim(C-n(1, 2, ... , t)) >= inverted right perpendicular2t/3inverted left perpendicular + 1 for t >= 5, where the equality holds if and only if t = 5 and n = 13. Thus dim(C-n(1, 2, ... , t)) >= inverted right perpendicular2t/3inverted left perpendicular + 2 for t >= 6. This bound 3 is sharp for every t >= 6.
