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SPLITTING NUMBER is NPcomplete
, 1997
"... We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such ope ..."
Abstract

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We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v 1 and v 2 , and attaches the neighbors of v either to v 1 or to v 2 . We prove that the splitting number decision problem is NPcomplete. We obtain as a consequence that planar subgraph remains NPcomplete when restricted to graphs with maximum degree 3, when restricted to graphs with no subdivision of K 5 , or when restricted to cubic graphs, problems that have been open since 1979.
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.
SPLITTING NUMBER is NPcomplete (Extended Abstract)
"... . We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such op ..."
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. We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v1 and v2 , and attaches the neighbors of v either to v1 or to v2 . We prove that the splitting number decision problem is NPcomplete when restricted to cubic graphs. We obtain as a consequence that planar subgraph remains NPcomplete when restricted to cubic graphs. Note that NPcompleteness for cubic graphs implies NPcompleteness for graphs not containing a subdivision of K5 as a subgraph. 1 Introduction Applications in Computer Science are frequently modeled with nonplanar graphs. Graph visualization and VLSI projects many times require strategies of layout techniques. Layout algorithms are limited to special classes of graphs. Fo...