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**1 - 2**of**2**### The Splitting Number and Skewness of . . .

"... The skewness of a graph G is the minimum number of edges that need to be deleted from G to produce a planar graph. The splitting number of a graph G is the minimum number of splitting steps needed to turn G into a planar graph; where each step replaces some of the edges fu; vg incident to a sele ..."

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The skewness of a graph G is the minimum number of edges that need to be deleted from G to produce a planar graph. The splitting number of a graph G is the minimum number of splitting steps needed to turn G into a planar graph; where each step replaces some of the edges fu; vg incident to a selected vertex u by edges fu ; vg, where u is a new vertex. We show that the splitting number of the toroidal grid graph Cn \Theta Cm is minfn; mg \Gamma 2ffi n;3 ffi m;3 \Gamma ffi n;4 ffi m;3 \Gamma ffi n;3 ffi m;4 and its skewness is minfn; mg \Gamma ffi n;3 ffi m;3 \Gamma ffi n;4 ffi m;3 \Gamma ffi n;3 ffi m;4 . Here, ffi is the Kronecker symbol, i.e., ffi i;j is 1 if i = j, and 0 if i 6= j.

### An Evolutionary Algorithm for Graph Planarisation by Vertex Deletion

"... Abstract: A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The proposed algorithm aims to delete a minimum number of vertices to achiev ..."

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Abstract: A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The proposed algorithm aims to delete a minimum number of vertices to achieve its goal. The vertex deletion number of a graph G = (V,E) is the smallest integer k ≥ 0 such that there is an induced planar subgraph of G obtained by the removal of k vertices of G. Considering that the corresponding decision problem is NP-complete and an approximation algorithm for graph planarisation by vertices deletion does not exist, this work proposes an evolutionary algorithm that uses a constructive heuristic algorithm to planarise a graph. This constructive heuristic has time complexity of O(n+m), where m = |V | and n = |E|, and it is based on the PQ-trees data structure and on the vertex deletion operation. The algorithm performance is verified by means of case studies. 1