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Edge intersection graphs of L-shaped paths in grids

Kathie Cameron, Steven Chaplick, Chính T. Hoàng
2016 Discrete Applied Mathematics  
That is, the edge intersection graphs where each vertex is represented by one of the following shapes: " , , and we consider zero bend paths (i.e., | and -) to be degenerate s.  ...  In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid.  ...  Suppose some vertex c ∈ C is such that all of its neighbors in S have degree one. Then G is an [ ]-graph iff G − c is. 2 Observation 13 Let G be a gem-free graph with a split partition (C, S).  ... 
doi:10.1016/j.dam.2015.01.039 fatcat:53msztfvvvhizfer3oqkleneea

Page 6363 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews  
classes of P;-free graphs.  ...  We give polynomial time algorithms for the prob- lem restricted to interval graphs, cographs, directed path graphs and partial k-trees.  ... 

MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling [article]

Tan Nhat Tran, Shuhei Tsujie
2022 arXiv   pre-print
It is known that the ideal and free graphic arrangements correspond to the unit interval and chordal graphs respectively.  ...  We prove that a graphic arrangement is MAT-free if and only if the underlying graph is strongly chordal.  ...  We thank Professor Takuro Abe for posing a question that whether or not strongly chordal graphs are closed under taking contraction which guides us to the rising sun example in §6(B).  ... 
arXiv:2204.08878v1 fatcat:e5h35w2h7jajvfugwb3eeq3w44

Semantic Word Cloud Representations: Hardness and Approximation Algorithms [chapter]

Lukas Barth, Sara Irina Fabrikant, Stephen G. Kobourov, Anna Lubiw, Martin Nöllenburg, Yoshio Okamoto, Sergey Pupyrev, Claudio Squarcella, Torsten Ueckerdt, Alexander Wolff
2014 Lecture Notes in Computer Science  
We study a geometric representation problem, where we are given a set B of axis-aligned rectangles (boxes) with fixed dimensions and a graph with vertex set B.  ...  The task is to place the rectangles without overlap such that two rectangles touch if the graph contains an edge between them. We call this problem CONTACT REPRESENTATION OF WORD NETWORKS (CROWN).  ...  MAX-CROWN on the class of graphs that can be partitioned in polynomial time into k star forests admits an α/k-approximation algorithm. Proof.  ... 
doi:10.1007/978-3-642-54423-1_45 fatcat:kfbf54o2lbfwzmrg46ypwbn4k4

Page 3677 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
” could also be called “crown-free”.  ...  The Johnson graph J = J(n,d) has as vertex set the collection of d- sets of an n-set X; two vertices y, db are adjacent whenever yM6 has size d— 1.  ... 

The class of (P_7,C_4,C_5)-free graphs: decomposition, algorithms, and χ-boundedness [article]

Kathie Cameron, Shenwei Huang, Irena Penev, Vaidy Sivaraman
2018 arXiv   pre-print
For a family H of graphs, we say that a graph G is H-free if no induced subgraph of G is isomorphic to any graph in H.  ...  We present a decomposition theorem for the class of (P_7,C_4,C_5)-free graphs; in fact, we give a complete structural characterization of (P_7,C_4,C_5)-free graphs that do not admit a clique-cutset.  ...  the class of H-free graphs.  ... 
arXiv:1803.03315v1 fatcat:253czpqwcfhe5pdit7zdohk2ry

Local Boxicity and Maximum Degree [article]

Atrayee Majumder, Rogers Mathew
2022 arXiv   pre-print
as a non-universal vertex in at most l of these interval graphs.  ...  The local boxicity of a graph G, denoted by lbox(G), is the minimum positive integer l such that G can be obtained using the intersection of k (, where k ≥ l,) interval graphs where each vertex of G appears  ...  A claw-free graph is a graph that contains no claw graph as its induced subgraph. We show that the local boxicity of a claw-free graph having a maximum degree of ∆ is at most 2∆.  ... 
arXiv:1810.02963v4 fatcat:dfyynsoh7jf2rnj5mxw3yiwixe

Tolerance Graphs [chapter]

Martin Golumbic
2013 Discrete Mathematics and Its Applications  
What is the complexity of recognizing interval probe graphs in the general case where no partition is given in advance? 8.  ...  NeST, 211, 217 core, 183 cotree, 74 coTT graph, 223, 232, 234, 235, 255 covering, 27 crown, 185 cutpoint, 243 cutset, 59 degree, 23 degree partition, 39 diameter, 203 diasteroidal triple,  ... 
doi:10.1201/b16132-66 fatcat:j2i3oyr2e5au7mtjsdmhmcaezm

A survey of the algorithmic aspects of modular decomposition

Michel Habib, Christophe Paul
2010 Computer Science Review  
A B S T R A C T Modular decomposition is a technique that applies to (but is not restricted to) graphs. The notion of a module naturally appears in the proofs of many graph theoretical theorems.  ...  This paper survey the ideas and techniques that arose from this line of research.  ...  The cographs are exactly the P 4 -free graphs (Fig. 5 ).  ... 
doi:10.1016/j.cosrev.2010.01.001 fatcat:5ylx2ohitbarpkkr3jatb3hqwu

Semantic Word Cloud Representations: Hardness and Approximation Algorithms [article]

Lukas Barth, Sara Irina Fabrikant, Stephen Kobourov, Anna Lubiw, Martin Nöllenburg, Yoshio Okamoto, Sergey Pupyrev, Claudio Squarcella, Torsten Ueckerdt, Alexander Wolff
2013 arXiv   pre-print
We study a geometric representation problem, where we are given a set R of axis-aligned rectangles with fixed dimensions and a graph with vertex set R.  ...  For this problem, we present constant-factor approximations for several graph classes, namely stars, trees, planar graphs, and graphs of bounded degree.  ...  5 Following the idea of Theorem 5, we can approximate MAX-CROWN by applying Lemma 1 to a partition of the supporting graph into sets of disjoint cycles.  ... 
arXiv:1311.4778v1 fatcat:fuy36i367jfxpcncyzqato2yzm

A survey on algorithmic aspects of modular decomposition [article]

Michel Habib, Christophe Paul
2009 arXiv   pre-print
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems.  ...  This paper survey the ideas and techniques that arose from this line of research.  ...  Partitioning the vertex set of a graph with respect to the neighbourhood of some vertex is a common operation in graph algorithms.  ... 
arXiv:0912.1457v2 fatcat:oqxqjjnoejgqrlr72plmzeisga

An approximation result for the interval coloring problem on claw-free chordal graphs

Giuseppe Confessore, Paolo Dell'Olmo, Stefano Giordani
2002 Discrete Applied Mathematics  
We focus our attention on claw-free chordal graphs, and show how to ÿnd an orientation of such a graph in linear time, which guarantees that each path is covered by at most two maximal cliques.  ...  This is equivalent to ÿnding a k-approximate solution for the interval coloring problem on a graph.  ...  For the subclass of crown-free interval graphs, Confessore et al.  ... 
doi:10.1016/s0166-218x(01)00282-7 fatcat:lpxsf75vone55nj27czevb3hda

Interval k-Graphs and Orders [article]

David E. Brown, Breeann M. Flesch, Larry J. Langley
2016 arXiv   pre-print
An interval k-graph is the intersection graph of a family I of intervals of the real line partitioned into at most k classes with vertices adjacent if and only if their corresponding intervals intersect  ...  In this paper we discuss the interval k-graphs that are the incomparability graphs of orders; i.e., cocomparability interval k-graphs or interval k-orders.  ...  A graph G is a probe interval graph if its vertices can be partitioned into sets P (probes) and N (nonprobes) with an interval of the real line corresponding to each vertex, and vertices adjacent if and  ... 
arXiv:1602.08669v1 fatcat:jrq5t2wplrhzfdod35bewm3xte

Directed Nowhere Dense Classes of Graphs [article]

Stephan Kreutzer, Siamak Tazari
2011 arXiv   pre-print
We then study in depth one type of classes of directed graphs which we call nowhere crownful.  ...  The classes are very general as they include, on one hand, all classes of directed graphs whose underlying undirected class is nowhere dense, such as planar, bounded-genus, and H-minor-free graphs; and  ...  . , (z −1 , z = k]} be a set of given intervals partitioning [1, k].  ... 
arXiv:1104.3808v1 fatcat:pqysgunz6bg3naxkjz53dpj3cm

Solving Problems on Generalized Convex Graphs via Mim-Width [article]

Flavia Bonomo-Braberman and Nick Brettell and Andrea Munaro and Daniël Paulusma
2022 arXiv   pre-print
Many 𝖭𝖯-complete problems, including problems such as Dominating Set, Feedback Vertex Set, Induced Matching and List k-Colouring, become polynomial-time solvable for ℋ-convex graphs when ℋ is the set  ...  A bipartite graph G=(A,B,E) is H-convex, for some family of graphs H, if there exists a graph H∈ H with V(H)=A such that the set of neighbours in A of each b∈ B induces a connected subgraph of H.  ...  It remains to observe that every chordal bipartite graph is (K 3 S 3 , K 3 K 3 )-free.  ... 
arXiv:2008.09004v3 fatcat:rp3f7zfadzbb7ipmw7q5yw5oym
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