Roman Domination Number of the Cartesian Products of Paths and Cycles

Abstract

Roman domination is an historically inspired variety of domination in graphs, in which vertices are assigned a value from the set $\{0,1,2\}$ in such a way that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. The Roman domination number is the minimum possible sum of all values in such an assignment. Using an algebraic approach we present an $O(C)$-time algorithm for computing the Roman domination numbers of special classes of graphs called polygraphs, which include rotagraphs and fasciagraphs. Using this algorithm we determine formulas for the Roman domination numbers of the Cartesian products of the form $P_n\Box P_k$, $P_n\Box C_k$, for $k\leq8$ and $n \in {\mathbb N}$, and $C_n\Box P_k$ and $C_n\Box C_k$, for $k\leq 6$ and $n \in {\mathbb N}$, for paths $P_n$ and cycles $C_n$. We also find all special graphs called Roman graphs in these families of graphs.

A General Lower Bound for the Domination Number of Cylindrical Graphs

“…We begin with the second type. Particular dominating sets of P m ✷C n , with m ≥ n, based on regular patterns, give that (see [19]) However, the comparison with the exact values given in [8], for the cases 16 ≤ m ≤ 22, leads us to think that the construction shown in [19] for the case n = 5k is optimal, for m big enough (see Figure 2). Clearly this construction can be done for 5 ≤ n ≡ 0 (mod 5) and any m ≥ 2 and it provides a dominating set with (m+2)n 5 = (m + 2)k vertices.…”

“…In the case n ≡ 0 (mod 5), the construction provided in [19], that we have shown in Figure 2, of a dominating set of P m ✷C 5k with (m + 2)k vertices, is likely to be the optimal construction. If this is true, the lower bound of Equation 4 is not tight in this case and following this intuition, we have conjectured how a better general lower bound should be.…”

“…In addition to Theorem 2, we need to apply the standard recurrence argument for the (min, +) matrix multiplication (see [12,13,16,19,21]). In the case r = 10, the computation of matrix A and its powers gives that (A 120 ) i,j = 1 + (A 119 ) i,j , for every i, j, that is, A 119+1 = 1 A 119 .…”

“…The final paper concerning this problem [12] closes a list of partial results regarding exact values for particular cases [1,5,7,15] and general upper and lower bounds [4,7,13], among other papers.Contrary to grids, the complete computation of the domination number of cylindrical graphs, that is, the Cartesian product of a path and a cycle, is still open. Currently, there are some partial results, see for instance [8,16,19], showing different techniques to compute exact values of particular cases, where one of the parameters, the path length or the cycle length, is small enough.We devote this paper to computing a general lower bound for the domination number of the cylindrical graphs, that is tight if the length of the cycle is a multiple of five. Our strategy to obtain this bound is similar to the approach in [13], that uses the concept of wasted domination.…”

“…Contrary to grids, the complete computation of the domination number of cylindrical graphs, that is, the Cartesian product of a path and a cycle, is still open. Currently, there are some partial results, see for instance [8,16,19], showing different techniques to compute exact values of particular cases, where one of the parameters, the path length or the cycle length, is small enough.…”