the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

Let~$G$ be a simple undirected graph. A broadcast on~$G$ is a function $f : V(G)\rightarrow\NNNNN$ such that $f(v)\le e_G(v)$ holds for every vertex~$v$ of~$G$, where $e_G(v)$ denotes the eccentricity of~$v$ in~$G$, is, maximum distance from~$v$ to any other vertex of~$G$. The cost of~$f$ is value $\cost(f)=\sum_{v\in V(G)}f(v)$. A broadcast~$f$ independent if two distinct vertices $u$ an...

After 2-crossing-critical graphs were characterized in 2016, their most general subfamily, large 3-connected graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for domination independence number.

we consider a dynamic domination problem for graphs in which an infinitesequence of attacks occur at vertices with guards and the guard at theattacked vertex is required to vacate the vertex by moving to a neighboringvertex with no guard. other guards are allowed to move at the same time, andbefore and after each attack and the resulting guard movements, the verticescontaining guards form a dom...