Browsing by Subject "tensor product"

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Access status: Open Access ,
On the existence of independent (1,k) -dominating sets for k∈{1,2} in two products of graphs
(Wydawnictwa AGH, 2026) Bednarz, Paweł; Michalski, Adrian; Paja, Natalia
A subset $J$ of vertices is said to be a $(1,k)$-dominating set if every vertex $v$ not belonging to the set $J$ has a neighbour in $J$ and there exists also another vertex in $J$ within the distance at most $k$ from $v$. In this paper, we study the problem of the existence of independent $(1,k)$-dominating sets for $k\in\{1,2\}$ in the tensor product and in the strong product of two graphs. We give complete characterisations of these graph products, which have independent $(1,1)$-dominating sets or independent $(1,2)$-dominating sets, with respect to the properties of their factors.
Access status: Open Access ,
Vulnerability parameters of tensor product of complete equipartite graphs
(2013) Paulraja, P.; Sheeba Agnes, V.
Let $G_1$ and $G_2$ be two simple graphs. The tensor product of $G_1$ and $G_2$, denoted by $G_{1}\times G_{2}$, has vertex set $V(G_{1}\times G_{2})=V(G_{1})\times V(G_{2})$ and edge set $E(G_{1}\times G_{2})=\{(u_{1},v_{1})(u_{2},v_{2}):u_{1}u_{2}\in E(G_{1})\}$. In this paper, we determine vulnerability parameters such as toughness, scattering number, integrity and tenacity of the tensor product of the graphs $K_{r(s)}\times K_{m(n)}$ for $r\geq 3, m\geq 3, s\geq 1$ and $n\geq 1,$ where $K_{r(s)}$ denotes the complete $r$-partite graph in which each part has s vertices. Using the results obtained here the theorems proved in [Aygul Mamut, Elkin Vumar, <i>Vertex Vulnerability Parameters of Kronecker Products of Complete Graphs</i>, Information Processing Letters 106 (2008), 258–262] are obtained as corollaries.