In this paper, we propose a polynomial-time algorithm to test whether a given graph contains a subdivision of K 4 as an induced subgraph.

It was conjectured by Hajós that graphs containing no K 5 -subdivision are 4-colorable. Previous results show any possible minimum counterexample to Hajós' conjecture, called graph, is 4-connected but not 5-connected. In this paper, we if a graph admits 4-cut or 5-cut with planar side then the must be small contains special wheel. This step in our effort reduce conjecture Four Color Theorem.

Reverse subdivision aims at constructing a coarser representation of an object given by a fine polygon mesh. In this paper, we first derive a mask for reverse Loop subdivision that can be applied to both regular and extraordinary vertices. The mask is parameterized, and thus can also be used in reversing variants of Loop subdivision, such as those proposed by Warren and Litke. We apply this mas...

Let H be a multigraph and G a graph containing a subgraph isomorphic to a subdivision of H, with S ⊂ V (G) (the ground set) the image of V (H) under the isomorphism. We consider connectivity and minimum degree or degree sum conditions sufficient to imply there is a spanning subgraph of G isomorphic to a subdivision of H on the same ground set. These results generalize a number of theorems in th...

In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of...

In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of...

Subdivision surfaces are de ned by a mesh of points and by one or more re nement rules according to which new subsets of points are substituted for existing subsets which are usually smaller in size, to yield re ned meshes. The re nement rules de ning a subdivision surface are known collectively as the re nement process de ning the surface. Re nement processes of interest are any for which the ...

Neutrosophic theory has many applications in graph theory, bipolar single valued neutrosophic graphs (BSVNGs) is the generalization of fuzzy graphs and intuitionistic fuzzy graphs, SVNGs. In this paper we introduce some types of BSVNGs, such as subdivision BSVNGs, middle BSVNGs, total BSVNGs and bipolar single valued neutrosophic line graphs (BSVNLGs), also investigate the isomorphism, co weak ...

We show that for every positive integer k, any tournament with minimum out-degree at least (2+o(1))k2 contains a subdivision of the complete directed graph on k vertices, where each path has length most 3. This result is best possible condition (up to multiplicative factor 8), and it tight respect paths. It may be viewed as analogue theorem proved by Bollobás Thomason, independently Komlós Szem...

We prove that the (d − 2)-nd barycentric subdivision of every convex d-ball is shellable. This yields a new characterization of the PL property in terms of shellability: A sphere or a ball is PL if and only if it becomes shellable after sufficiently many barycentric subdivisions. This improves results by Whitehead, Zeeman and Glaser. Moreover, we show the Zeeman conjecture is equivalent to the ...