Signed total double Roman dominatıon numbers in digraphs

Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) P x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consist of all in-neighbors of v, and (ii) if f(v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x∈V (D) f(x). The signed total double Roman domination number (STDRD-number) γtsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for γtsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.Publisher's Versio

The Signed Domination Number of Cartesian Products of Directed Cycles

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function is called a signed dominating function (SDF) iffor each vertex v. The weight w(f ) of f is defined by. The signed domination number of a digraph D is gs(D) = min{w(f ) : f is an SDF of D}. Let Cmn denotes the Cartesian product of directed cycles of length m and n. In this paper, we determine the exact value of signed domination number of some classes of Cartesian product of directed cycles. In particular, we prove that: (a) gs(C3n) = n if n 0(mod 3), otherwise gs(C3n) = n + 2. (b) gs(C4n) = 2n. (c) gs(C5n) = 2n if n 0(mod 10), gs(C5n) = 2n + 1 if n 3, 5, 7(mod 10), gs(C5n) = 2n + 2 if n 2, 4, 6, 8(mod 10), gs(C5n) = 2n + 3 if n 1,9(mod 10). (d) gs(C6n) = 2n if n 0(mod 3), otherwise gs(C6n) = 2n + 4. (e) gs(C7n) = 3n

The signed (k,k) -domatic number of digraphs

et $D$ be a finite and simple digraph with vertex set $V(D)$, and let $f:V(D)rightarrow{-1,1}$ be a two-valued function. If $kge 1$ is an integer and $sum_{xin N^-[v]}f(x)ge k$ for each $vin V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from which arcs go into $v$, then $f$ is a signed $k$-dominating function on $D$. A set ${f_1,f_2,ldots,f_d}$ of distinct signed $k$-dominating functions on $D$ with the property that $sum_{i=1}^df_i(x)le k$ for each $xin V(D)$, is called a signed $(k,k)$-dominating family (of functions) on $D$. The maximum number of functions in a signed $(k,k)$-dominating family on $D$ is the signed $(k,k)$-domatic number on $D$, denoted by $d_{S}^{k}(D)$. In this paper, we initiate the study of the signed $(k,k)$-domatic number of digraphs, and we present different bounds on $d_{S}^{k}(D)$. Some of our results are extensions of well-known properties of the signed domatic number $d_S(D)=d_{S}^{1}(D)$ of digraphs $D$ as well as the signed $(k,k)$-domatic number $d_S^k(G)$ of graphs $G$

Signed double Roman domination on cubic graphs

The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one $3$-labeled neighbor or at least two $2$-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 $n$ for which we present a sharp $n/2+\Theta(1)$ 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 $2\times m$ grid graphs, among other results

α-Domination

AbstractLet G=(V,E) be any graph with n vertices, m edges and no isolated vertices. For some α with 0<α⩽1 and a set S⊆V, we say that S is α-dominating if for all v∈V−S,|N(v)∩S|⩾α|N(v)|. The size of a smallest such S is called the α-domination number and is denoted by γα(G). In this paper, we introduce α-domination, discuss bounds for γ1/2(G) for the King's graph, and give bounds for γα(G) for a general α, 0<α⩽1. Furthermore, we show that the problem of deciding whether γα(G)⩽k is NP-complete