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Edge and vertex intersection of paths in a tree

Martin Charles Golumbic, Robert E. Jamison
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

Fǎnicǎ Gavril
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]

Chenyu Yan, Yang Xiang, Feodor F. Dragan
2009 Lecture Notes in Computer Science
We prove that this routing labeling scheme has 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

Maciej M. Sysło
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

Martin Charles Golumbic, Marina Lipshteyn, Michal Stern
2008 Discrete Applied Mathematics
We consider 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

Guantao Chen, Julia Ehrenmüller, Cristina G. Fernandes, Carl Georg Heise, Songling Shan, Ping Yang, Amy N. Yates
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]

Tim Dwyer, Lev Nachmanson
2010 Lecture Notes in Computer Science
Recent work 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]

Blake Mellor
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]

Arman Boyacı, Tınaz Ekim, Mordechai Shalom, Shmuel Zaks
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

Binay K. Bhattacharya, Subir Kumar Ghosh
2001 Computational geometry
Using our characterization 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.  ...

BLAKE MELLOR
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

Martin Charles Golumbic, Marina Lipshteyn, Michal Stern
2008 Discrete Applied Mathematics
Jamison, 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]

Dekel Tsur
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

Martin Charles Golumbic, Marina Lipshteyn, Michal Stern
2008 Discrete Mathematics
An undirected graph G is called an 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

Chenyu Yan, Yang Xiang, Feodor F. Dragan
2012 Computational geometry
x 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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