Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a $H$ treewidth at most 4 path $P$ such $G\subseteq H \boxtimes P K_{\max\{2g,3\}}$. We improve this result by replacing "4" "3" planar. in fact prove more general terms of so-called framed graphs. This implies $(g,d)$-map contained $ P\boxtimes K_\ell$, some planar $3$...

Let G = (V, E) be a simple connected graph. The geodesic distance between two vertices u, v in is the length terms of number edges shortest path vertices. In this paper, we determine Gd-distance wreath product graphs. Using formulae obtained here, have found exact value some classes

The rupture degree of a noncomplete-connected graph G is defined by , where the number components and order largest component of. In this paper, we determine some Cartesian product graphs.