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## Filters

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Edge and vertex intersection of paths in a tree

1985
*
Discrete Mathematics
*

*In*this paper we continue the investigation

*of*the class

*of*

*edge*

*intersection*graphs

*of*

*a*collection

*of*

*paths*

*in*

*a*

*tree*(EPT graphs) where two

*paths*

*edge*

*intersect*if they share an

*edge*. ... The class

*of*EPT graphs differs from the class known as

*path*graphs, the latter being the class

*of*

*vertex*

*intersection*graphs

*of*

*paths*

*in*

*a*

*tree*. ... Fig. 1 shows

*a*collection

*of*5

*paths*

*and*their EPT graphs. Note that if two

*paths*

*intersect*

*in*

*a*single

*vertex*(

*vertex*

*intersect*), they do not

*edge*

*intersect*. Remark. ...

##
###
Intersection graphs of Helly families of subtrees

1996
*
Discrete Applied Mathematics
*

*In*addition, we describe polynomial-time recognition algorithms for the

*intersection*graphs

*and*for the perfect

*intersection*graphs

*of*Helly families

*of*subtrees

*in*cacti graphs. ...

*In*the present paper we prove that every graph is an

*intersection*graph

*of*

*a*Helly family

*of*subtrees

*of*

*a*graph without triangles. ... We divide every

*edge*

*of*CY (

*and*

*of*every

*path*

*of*S) into two

*edges*by inserting

*a*new

*vertex*

*in*its middle; we denote the new cycle by CY'. ...

##
###
Compact and Low Delay Routing Labeling Scheme for Unit Disk Graphs
[chapter]

2009
*
Lecture Notes in Computer Science
*

We prove that this routing labeling scheme has

doi:10.1007/978-3-642-03367-4_49
fatcat:jxxwvp6qwvgjlec2iunrsehfg4
*a*constant hop route-stretch (= hop delay), i.e., for each two vertices x*and*y*of*G, it produces*a*routing*path*with h(x, y) hops (*edges*) such that h(x, ... We show that one can assign each*vertex**of*an n-*vertex*UDG G*a*compact O(log 2 n)-bit label such that, given the label*of**a*source*vertex**and*the label*of**a*destination, it is possible to compute efficiently ...*In*what follows, we will transform*tree*T orig into*a*special spanning*tree*T for the planar graph There are two other kinds*of**edge**intersections**in*G: the*intersection*between*a**tree*-*edge**and**a*non-*tree*-*edge*...##
###
Triangulated edge intersection graphs of paths in a tree

1985
*
Discrete Mathematics
*

*in*T,

*and*G is called an

*edge*

*intersection*graph

*of*

*paths*

*in*

*a*

*tree*(EPT graph) if G = O(E(T), 9) for some

*path*collection 9

*in*

*a*

*tree*T. ...

*A*graph G is

*a*

*vertex*

*intersection*graph

*of*

*paths*

*in*

*a*

*tree*(shortly, VPT graph) if G = f2( V(T), 9) for

*a*certain

*tree*T

*and*

*a*

*path*collection 9? ... Golumbic

*and*to the referees for their suggestions on improving the presentation

*of*this paper. ...

##
###
The k-edge intersection graphs of paths in a tree

2008
*
Discrete Applied Mathematics
*

We consider

doi:10.1016/j.dam.2006.06.019
fatcat:nfjkjpdyzvcdlkqgc5tcnv43ka
*a*generalization*of**edge**intersection*graphs*of**paths**in**a**tree*. Let P be*a*collection*of*nontrivial simple*paths**in**a**tree*T . ... An undirected graph G is called*a*k-*edge**intersection*graph*of**paths**in**a**tree*,*and*denoted by k-EPT, if G = k (P) for some P*and*T . ... An undirected graph G is called*a**vertex**intersection*graph*of**paths**in**a**tree*(VPT) if G = (P) for some P*and*T . We define P, T to be*a*VPT representation*of*G. ...##
###
Nonempty intersection of longest paths in series–parallel graphs

2017
*
Discrete Mathematics
*

*In*1966 Gallai asked whether all longest

*paths*

*in*

*a*connected graph have nonempty

*intersection*. This is not true

*in*general

*and*various counterexamples have been found. ... We present

*a*proof that every connected series-parallel graph has

*a*

*vertex*that is common to all

*of*its longest

*paths*. ...

*A*

*vertex*

*of*attachment

*of*H

*in*G is

*a*

*vertex*

*of*H that is incident to some

*edge*

*of*G that is not an

*edge*

*of*H. Let T be

*a*2-

*tree*

*and*let x

*and*y be two adjacent vertices

*in*T . ...

##
###
Fast Edge-Routing for Large Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

Recent work

doi:10.1007/978-3-642-11805-0_15
fatcat:cuzisrgocvet5agxeacuczzqqm
*in*this area involves finding shortest*paths**in**a*tangent-visibility graph. ... To produce high quality drawings*of*graphs with nodes drawn as shapes it is important to find routes for the*edges*which do not*intersect*node boundaries. ...*In*shortcutting we try to skip each internal*vertex**of*the shortest*path*by removing it*and*checking that the*path*still does not*intersect*the interior*of*an obstacle. ...##
###
Tree Diagrams for String Links
[article]

2005
*
arXiv
*
pre-print

*In*previous work, the author defined the

*intersection*graph

*of*

*a*chord diagram associated with string links (as

*in*the theory

*of*finite type invariants). ...

*In*this paper, we classify the

*trees*which can be obtained as

*intersection*graphs

*of*string link diagrams. ... Definition 4

*A*bough

*of*

*a*

*vertex*v

*in*

*a*marked

*tree*T is

*a*connected component

*of*T \v (the graph which results from removing the

*vertex*v

*and*all

*edges*incident to v). ...

##
###
Graphs of Edge-Intersecting Non-splitting Paths in a Tree: Towards Hole Representations
[chapter]

2013
*
Lecture Notes in Computer Science
*

*In*this work, we define the graph Enpt(P)

*of*

*edge*

*intersecting*non-splitting

*paths*

*of*

*a*

*tree*, termed the ENPT graph, as the (

*edge*) graph having

*a*

*vertex*for each

*path*

*in*P,

*and*an

*edge*between every pair ... Given

*a*

*tree*

*and*

*a*set P

*of*non-trivial simple

*paths*on it, Vpt(P) is the VPT graph (i.e. the

*vertex*

*intersection*graph)

*of*P,

*and*Ept(P) is the EPT graph (i.e. the

*edge*

*intersection*graph)

*of*the

*paths*... Introduction Background Given

*a*

*tree*T

*and*

*a*set P

*of*non-trivial simple

*paths*

*in*T , the

*Vertex*

*Intersection*Graph

*of*

*Paths*

*in*

*a*

*Tree*(VPT)

*and*the

*Edge*

*Intersection*Graph

*of*

*Paths*

*in*

*a*

*Tree*(EPT)

*of*...

##
###
Characterizing LR-visibility polygons and related problems☆☆The extended abstract of this paper was presented at the Tenth Canadian Conference on Computational Geometry, 1998

2001
*
Computational geometry
*

Using our characterization

doi:10.1016/s0925-7721(00)00024-9
fatcat:a5ktlbhvcnfildru3vbdk2jwgi
*of*LR-visibility polygons, we show that the shortest*path**tree*rooted at*a**vertex*or*a*boundary point can be computed*in*linear time for*a*class*of*polygons which contains LR-visibility ...*In*this paper we derive properties*of*shortest*paths**in*LR-visibility polygons*and*present*a*characterization*of*LR visibility polygons*in*terms*of*shortest*paths*between vertices. ... Acknowledgements We gratefully acknowledge the helpful comments*of*Piyush Kumar, Sudeb Pal*and*Tom Shermer. The authors thank anonymous referees for suggesting improvements to the original paper. ...##
###
TREE DIAGRAMS FOR STRING LINKS

2006
*
Journal of knot theory and its ramifications
*

*In*previous work, the author defined the

*intersection*graph

*of*

*a*chord diagram associated with

*a*string link (as

*in*the theory

*of*finite type invariants). ...

*In*this paper, we classify the

*trees*which can be obtained as

*intersection*graphs

*of*string link diagrams. ... Definition 4

*A*bough

*of*

*a*

*vertex*v

*in*

*a*marked

*tree*T is

*a*connected component

*of*T \v (the graph which results from removing the

*vertex*v

*and*all

*edges*incident to v). ...

##
###
Equivalences and the complete hierarchy of intersection graphs of paths in a tree

2008
*
Discrete Applied Mathematics
*

Jamison,

doi:10.1016/j.dam.2008.05.019
fatcat:22in53ymrnd4lemk73i2gcqo6y
*Edge**and**vertex**intersection**of**paths**in**a**tree*, Discrete Math. 55 (1985) 151-159],*and*[h, 2, 2] graphs are known as the EPT graphs. ...*In*this paper, we investigate the class*of*[h, 2, t] graphs, i.e., the*intersection*graphs*of**paths**in**a**tree*. The [h, 2, 1] graphs are also known as*path*graphs [F. ... We also would like to find the relationship between orthodox k-EPT graphs restricted to degree 3*trees**and*other families*of*graphs*in*the hierarchy. ...##
###
l-path vertex cover is easier than l-hitting set for small l
[article]

2019
*
arXiv
*
pre-print

*In*the l-

*path*

*vertex*cover problem the input is an undirected graph G

*and*an integer k. ... The goal is to decide whether there is

*a*set

*of*vertices S

*of*size at most k such that G-S does not contain

*a*

*path*with l vertices. ... Note that the label

*of*an

*edge*(x, x i )

*in*the

*tree*T LPVCalg (G, k) is equal to the sum

*of*the labels

*of*the

*edges*on the

*path*from x to x i

*in*the

*tree*T(G, k). ...

##
###
Representing edge intersection graphs of paths on degree 4 trees

2008
*
Discrete Mathematics
*

An undirected graph G is called an

doi:10.1016/j.disc.2007.07.064
fatcat:wb4kv6xqtzhxdlig45zizcmswy
*edge**intersection*graph*of**paths**in**a**tree*if G = EPT(P) for some P*and*T. The EPT graphs are useful*in*network applications. ... The*edge**intersection*graph*of*P, denoted by EPT(P), has*vertex*set that corresponds to the members*of*P,*and*two vertices are joined by an*edge*if*and*only if the corresponding members*of*P share at least ... An undirected graph G is called*a**vertex**intersection*graph*of**paths**in**a**tree*(VPT) if G = VPT(P) for some P*and*T,*and*we call P, T*a*VPT representation*of*G. ...##
###
Compact and low delay routing labeling scheme for Unit Disk Graphs

2012
*
Computational geometry
*

x

doi:10.1016/j.comgeo.2012.01.015
fatcat:evpu75tdtjdbboxb27drwe6azy
*and*y*of*G, there exists*a**tree*T*in*T (G) with d T (x, y) 3 • d G (x, y) + 12. ... We prove that this routing labeling scheme has*a*constant hop routestretch (= hop delay), i.e., for each two vertices x*and*y*of*G, it produces*a*routing*path*with h(x, y) hops (*edges*) such that h(x, y ... Acknowledgements We would like to thank the anonymous referees for many useful suggestions*and*comments. ...
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