1-planarity of complete multipartite graphs

Czap, Július; Hudak, David E.

doi:10.1016/j.dam.2011.11.014

Cited by 33 publications

(32 citation statements)

References 10 publications

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“…If G is maximally dense then it is also maximal, but not vice versa. We remark that maximally dense graphs are sometimes called "optimal" in the literature (see, e.g., [14,17,39]). The following property holds.…”

Section: Propertymentioning

confidence: 99%

Fan-planarity: Properties and complexity

Binucci

Giacomo

Didimo

et al. 2015

Theoretical Computer Science

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In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt [35], who proved that every n-vertex fan-planar drawing has at most 5. n-. 10 edges, and that this bound is tight for n≥. 20. We extend their result from both the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and k-planarity. Also, we prove that testing fan-planarity in the variable embedding setting is NP-complete

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Section: Propertymentioning

confidence: 99%

Fan-planarity: Properties and complexity

Binucci

Giacomo

Didimo

et al. 2015

Theoretical Computer Science

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“…Observe that the complete bipartite graph K m,n is a subgraph of G + H. In the following, we will use the characterization of 1-planar complete multipartite graphs from [4]; the results are contained in Table 1, where the notation of sizes of parts of vertices is the following: a − b means the set a, b ∩ Z (the interval of integers); a− means all integers greater or equal to a. k = 2 K 1−,1 ; K 2−,2 ; K 3−6,3 ; K 4,4 k = 3 K 1−,1,1 ; K 2−6,2,1 ; K 2−4,2,2 ; K 3,3,1 k = 4 K 1−6,1,1,1 ; K 2−3,2,1,1 ; K 2,2,2,1−2 k = 5 K 1−2,1−2,1,1,1 k = 6 K 1,1,1,1,1,1 These results imply that G + H is not 1-planar if m ≥ 5, n ≥ 4 or m ≥ 7, n ≥ 3.…”

Section: The Join G + H With |V (G)| ≥ |V (H)| ≥mentioning

confidence: 99%

Joins of 1-planar graphs

Czap

Hudak

Madaras

2014

Acta. Math. Sin.-English Ser.

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A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join $G+H$ is 1-planar if and only if the pair $[G,H]$ is subgraph-majorized (that is, both $G$ and $H$ are subgraphs of graphs of the major pair) by one of pairs $[C_3 \cup C_3,C_3], [C_4,C_4], [C_4,C_3], [K_{2,1,1},P_3]$ in the case when both factors of the graph join have at least three vertices. If one factor has at most two vertices, then we give several necessary/sufficient conditions for the bigger factor

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“…Among complete multipartite graphs G, the planar ones are: K 2,m ; K 1,1,m ; K 1,2,2 ; K 1,1,1,1 = K 4 and their subgraphs. The 1-planar G are, besides above: K 6 ; K 1,1,1,6 ; K 1,1,2,3 ; K 2,2,2,2 ; K 1,1,1,2,2 and their subgraphs ( [11]) Given sets A 1 , . .…”

Section: 5mentioning

confidence: 99%

Generalized cut and metric polytopes of graphs and simplicial complexes

Deza¹,

Sikirić

2018

Optim Lett

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Given a graph G one can define the cut polytope CUTP(G) and the metric polytope METP(G) of this graph and those polytopes encode in a nice way the metric on the graph. According to Seymour's theorem, CUTP(G) = METP(G) if and only if K 5 is not a minor of G.We consider possibly extensions of this framework: (1) We compute the CUTP(G) and METP(G) for many graphs.(2) We define the oriented cut polytope WOMCUTP(G) and oriented multicut polytope OMCUTP(G) as well as their oriented metric version QMETP(G) and WQMETP(G). (3) We define an n-dimensional generalization of metric on simplicial complexes.

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1-planarity of complete multipartite graphs

Cited by 33 publications

References 10 publications

Fan-planarity: Properties and complexity

Fan-planarity: Properties and complexity

Joins of 1-planar graphs

Generalized cut and metric polytopes of graphs and simplicial complexes

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