On graph thickness, geometric thickness, and separator theorems (original) (raw)
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Geometric Thickness for the General Graphs
In this method the user gives a graph as input which is then processed. After processing the input graph is determined whether it is a planar graph or not. If the input graph is planar then a planar embedding of the given input graph is shown as the output having only one color. If the input graph is non planar then n layers of the given input graph is displayed where each layer is of different colors. Then each layer is also called as the Geometric-Thickness. The geometric thickness θ(G) of a graph G is the smallest integer t such that there exist a straight-line drawing of G and a partition of its straight-line edges into t subsets, where each subset induces a planar drawing. Over a decade ago, Hutchinson, Shermer, and Vince proved that any n-vertex graph with geometric thickness two can have at most 6n − 18 edges, and for every n ≥ 8 they constructed a geometric thickness two graph with 6n − 20 edge, but we taken the 6n-18 edges. And also we do the NP-hardness of coloring graphs of geometric thickness.
Remarks on the thickness and outerthickness of a graph
Computers & Mathematics with Applications, 2005
The thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed The thickness of complete bipartite graphs K,~,,~ is known for almost all values of m and n In this paper, we solve the thickness of complete bipartite graphs for unknown cases m < 30, rn <_ n. The new solutions coincide with the general formula and they were obtained by using a mmulated annealing algorithm
Thickness and outerthickness for embedded graphs
Discrete Mathematics, 2018
We consider the thickness θ(G) and outerthickness θ o (G) of a graph G in terms of its orientable and nonorientable genus. Dean and Hutchinson provided upper bounds for thickness of graphs in terms of their orientable genus. More recently, Concalves proved that the outerthickness of any planar graph is at most 2. In this paper, we apply the method of deleting spanning disks of embeddings to approximate the thickness and outerthickness of graphs. We first obtain better upper bounds for thickness. We then use a similar approach to provide upper bounds for outerthickness of graphs in terms of their orientable and nonorientable genera. Finally we show that the outerthickness of the torus (the maximum outerthickness of all toroidal graphs) is 3. We also show that all graphs embeddable in the double torus have thickness at most 3 and outerthickness at most 5.
On the fold thickness of graphs
Arabian Journal of Mathematics
The graph G'G′obtainedfromagraphGbyidentifyingtwononadjacentverticesinGhavingatleastonecommonneighboriscalleda1−foldofG.AsequenceG ′ obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor is called a 1-fold of G. A sequenceG′obtainedfromagraphGbyidentifyingtwononadjacentverticesinGhavingatleastonecommonneighboriscalleda1−foldofG.AsequenceG_0, G_1, G_2, \ldots , G_kG0,G1,G2,…,GkofgraphssuchthatG 0 , G 1 , G 2 , … , G k of graphs such thatG0,G1,G2,…,GkofgraphssuchthatG_0=GG0=GandG 0 = G andG0=GandG_iGiisa1−foldofG i is a 1-fold ofGiisa1−foldofG_{i-1}Gi−1foreachG i - 1 for eachGi−1foreachi=1, 2, \ldots , ki=1,2,…,kiscalledauniformk−foldingofGifthegraphsinthesequenceareallsingularorallnonsingular.ThefoldthicknessofGisthelargestkforwhichthereisauniformk−foldingofG.Weshowherethatthefoldthicknessofasingularbipartitegraphofordernisi = 1 , 2 , … , k is called a uniform k-folding of G if the graphs in the sequence are all singular or all nonsingular. The fold thickness of G is the largest k for which there is a uniform k-folding of G. We show here that the fold thickness of a singular bipartite graph of order n isi=1,2,…,kiscalledauniformk−foldingofGifthegraphsinthesequenceareallsingularorallnonsingular.ThefoldthicknessofGisthelargestkforwhichthereisauniformk−foldingofG.Weshowherethatthefoldthicknessofasingularbipartitegraphofordernisn-3$$ n - 3 . Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these fam...
A construction of thickness-minimal graphs
Discrete Mathematics, 1987
The thickness of a graph G is the minimum number of planar subgraphs whose union is G. A t-minimal graph is a graph of thickness t which contains no proper subgraph of thickness t. For each t ~> 2 we present an explicit construction of an infinite number of t-minimal graphs with connectivity 2, edge connectivity t, and minimum valency t.
Journal of Combinatorial Theory, Series B, 1979
The book thickness bt(G) of a graph G is defined, its basic properties are delineated, and relations are given with other invariants such as thickness, genus, and chromatic number. A graph G has book thickness bt(G) < 2 if and only if it is a subgraph of a hamiltonian planar graph, but we conjecture that there are planar graphs with arbitrarily high book thickness.
The Linear 2-Arboricity of Planar Graphs
Graphs and Combinatorics, 2003
Let G be a planar graph with maximum degree D and girth g. The linear 2-arboricity la 2 ðGÞ of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. We prove that (1) la 2 ðGÞ dðD þ 1Þ=2e þ 12; (2) la 2 ðGÞ dðD þ 1Þ=2e þ 6 if g ! 4; (3) la 2 ðGÞ dðD þ 1Þ=2e þ 2 if g ! 5; (4) la 2 ðGÞ dðD þ 1Þ=2e þ 1 if g ! 7.