The cardinality of a maximum minimal dominating set of a graph is called its upper domination number. The problem of computing this number is generally NP-hard but can be solved in polynomial time in some restricted graph classes. In this work, we consider the complexity and approximability of the weighted version of the problem in two special graph classes: planar bipartite, split. We also pro...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and rainbow-connected there a between each pair its vertices. The minimum number colors needed to rainbow-connect G connection G, denoted by rc(G). simple way color spanning tree with distinct then re-use any these remaining G. This proves that rc(G)≤|V(G)|−1. We ask whether stronger tree-like structures colo...

We investigate experimentally the Domatic Partition (DP) problem, the Independent Domatic Partition (IDP) problem and the Idomatic partition problem in Random Geometric Graphs (RGGs). In particular, we model these problems as Integer Linear Programs (ILPs), solve them optimally, and show on a large set of samples that RGGs are independent domatically full most likely (over 93% of the cases) and...

In this paper, we give upper bounds on the upper signed domination number of [l, k] graphs, which generalize some results obtained in other papers. Further, good lower bounds are established for the minus ksubdomination number γ−101 ks and signed k-subdomination number γ −11 ks .

Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K) of the knot, which is s(K) ≤ 2c(K). Furthermore, McCabe proved that s(K) ≤ c(K) + 3 for a 2-bridge knot or link, except in the cases of the unlink and the Hopf link. In this paper we construct any 2-bridge knot or link K of at least six crossings by using only c(K) + 2 straig...