1,386 research outputs found

    [1,2]-Domination in Generalized Petersen Graphs

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    A vertex subset SS of a graph G=(V,E)G=(V,E) is a [1,2][1,2]-dominating set if each vertex of V\SV\backslash S is adjacent to either one or two vertices in SS. The minimum cardinality of a [1,2][1,2]-dominating set of GG, denoted by γ[1,2](G)\gamma_{[1,2]}(G), is called the [1,2][1,2]-domination number of GG. In this paper the [1,2][1,2]-domination and the [1,2][1,2]-total domination numbers of the generalized Petersen graphs P(n,2)P(n,2) are determined

    Signed double Roman domination on cubic graphs

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    The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from {±1,2,3}\{\pm{}1,2,3\} to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled ±1\pm{}1 have at least one neighbor with label in {2,3}\{2,3\}; (ii) each vertex labeled −1-1 has one 33-labeled neighbor or at least two 22-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order nn for which we present a sharp n/2+Θ(1)n/2+\Theta(1) lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic 2×m2\times m grid graphs, among other results

    k-Tuple_Total_Domination_in_Inflated_Graphs

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    The inflated graph GIG_{I} of a graph GG with n(G)n(G) vertices is obtained from GG by replacing every vertex of degree dd of GG by a clique, which is isomorph to the complete graph KdK_{d}, and each edge (xi,xj)(x_{i},x_{j}) of GG is replaced by an edge (u,v)(u,v) in such a way that u∈Xiu\in X_{i}, v∈Xjv\in X_{j}, and two different edges of GG are replaced by non-adjacent edges of GIG_{I}. For integer k≥1k\geq 1, the kk-tuple total domination number γ×k,t(G)\gamma_{\times k,t}(G) of GG is the minimum cardinality of a kk-tuple total dominating set of GG, which is a set of vertices in GG such that every vertex of GG is adjacent to at least kk vertices in it. For existing this number, must the minimum degree of GG is at least kk. Here, we study the kk-tuple total domination number in inflated graphs when k≥2k\geq 2. First we prove that n(G)k≤γ×k,t(GI)≤n(G)(k+1)−1n(G)k\leq \gamma_{\times k,t}(G_{I})\leq n(G)(k+1)-1, and then we characterize graphs GG that the kk-tuple total domination number number of GIG_I is n(G)kn(G)k or n(G)k+1n(G)k+1. Then we find bounds for this number in the inflated graph GIG_I, when GG has a cut-edge ee or cut-vertex vv, in terms on the kk-tuple total domination number of the inflated graphs of the components of G−eG-e or vv-components of G−vG-v, respectively. Finally, we calculate this number in the inflated graphs that have obtained by some of the known graphs

    Rainbow domination and related problems on some classes of perfect graphs

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    Let k∈Nk \in \mathbb{N} and let GG be a graph. A function f:V(G)→2[k]f: V(G) \rightarrow 2^{[k]} is a rainbow function if, for every vertex xx with f(x)=∅f(x)=\emptyset, f(N(x))=[k]f(N(x)) =[k]. The rainbow domination number γkr(G)\gamma_{kr}(G) is the minimum of ∑x∈V(G)∣f(x)∣\sum_{x \in V(G)} |f(x)| over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs

    Power domination on triangular grids

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    The concept of power domination emerged from the problem of monitoring electrical systems. Given a graph G and a set S ⊆\subseteq V (G), a set M of monitored vertices is built as follows: at first, M contains only the vertices of S and their direct neighbors, and then each time a vertex in M has exactly one neighbor not in M, this neighbor is added to M. The power domination number of a graph G is the minimum size of a set S such that this process ends up with the set M containing every vertex of G. We here show that the power domination number of a triangular grid T\_k with hexagonal-shape border of length k -- 1 is exactly $\lceil k/3 \rceil.Comment: Canadian Conference on Computational Geometry, Jul 2017, Ottawa, Canad
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