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Homogeneously representable interval graphs

Dale Skrien, John Gimbel
1985 Discrete Mathematics  
We characterize those interval graphs G with the property that, for every vertex u, there exists an interval represention of G in which the interval representing 21 is the left-most (or right-most) interval  ...  in the representation. 0012-365)3/85/$3.30 0 1985, Elsevier Science Publishers B.V.  ...  For example, the graph in Fig. 1 is a homogeneously representable interval graph, but neither of the graphs in Fig. 2 is homogeneously representable.  ... 
doi:10.1016/0012-365x(85)90049-4 fatcat:6vu2wvfzive2pmtiwiyjpiuuye

On Homogeneously Representable Interval Graphs

Marcin Krzywkowski, Jerzy Topp
2015 National academy science letters  
We determine all graphs whose line graphs (middle graphs, total graphs, respectively) are homogeneously representable interval graphs.  ...  A graph G is called a homogeneously representable interval graph (shortly, an HRI graph) if for every vertex v of G there exists an interval representation of G in which the interval representing v is  ...  In the next two theorems we characterize all graphs G whose line graphs LðGÞ are homogeneously representable interval graphs.  ... 
doi:10.1007/s40009-015-0397-x fatcat:e2ojbqz4sfckdd6pukap7bvcay

On the Recognition of Fuzzy Circular Interval Graphs [article]

Gianpaolo Oriolo, Ugo Pietropaoli, Gautier Stauffer
2011 arXiv   pre-print
Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs.  ...  In this paper, we provide a polynomial-time algorithm for recognizing such graphs, and more importantly for building a suitable representation.  ...  G is a fuzzy linear interval graph if and only if G| (K 1 ,K 2 ) is a fuzzy linear interval graph and, from a representation for G, one may build in O(n 2 )-time a representation for G| (K 1 ,K 2 ) , and  ... 
arXiv:1104.2251v1 fatcat:32pzpopln5esrcydycsn5iaqxm

End vertices in interval graphs

John Gimbel
1988 Discrete Applied Mathematics  
We also present a short proof of a characterization of homogeneously representable interval graphs. <  ...  We characterize those vertices v in interval graphs for which there is some representation of G where the interval corresponding to v is an end interval.  ...  It is clear that no homogeneously representable interval graph contains either the S-path or the A-graph as induced subgraphs.  ... 
doi:10.1016/0166-218x(88)90071-6 fatcat:wokmaa7mgjgupeoqmsdiqyfuky

On sources in comparability graphs, with applications

S. Olariu
1992 Discrete Mathematics  
We characterize sources in comparability graphs and show that our result provides a unifying look at two recent results about interval graphs.  ...  An orientation 0 of a graph G is obtained by assigning unique directions to its  ...  Theorem 1 implies the following characterization of the homogeneously representable interval graphs. Corollary 1.2 (Skrien and Gimbel [6] ).  ... 
doi:10.1016/0012-365x(92)90721-q fatcat:nctlhvz7tjdtbcgubbfbkb7k7q

Page 4980 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
The authors prove that an interval graph G is a homogeneously representable interval graph if and only if  ...  An interval graph G is said to be homogeneously representable if for every vertex v in G there exists an interval representation of G in which the interval representing v is an end interval.  ... 

Interval graphs and related topics

Martin Charles Golumbic
1985 Discrete Mathematics  
Gimble, Homogeneously representable interval graphs, Discrete Math. 55 (1985) (this issue). [ 111 M.H.  ...  Skrien and Gimble [lo] call an interval graph G homogeneously representable if for every vertex u there exists an interval representation of G in which the interval representing n is the left-most (or  ... 
doi:10.1016/0012-365x(85)90039-1 fatcat:3lkhvtmp3jdv5drohuptcyj3su

Page 1062 of Mathematical Reviews Vol. 42, Issue 5 [page]

1971 Mathematical Reviews  
The graphs G which admit intersection representations where the family F is a family of intervals in a linearly ordered set have been characterized by several authors. A. Tucker [Bull. Amer. Math.  ...  Conditions for pseudo L,(k) graphs to be representation of a graph @ by arcs in a graph H. Such a geometrizable.  ... 

Twin-width and Limits of Tractability of FO Model Checking on Geometric Graphs [article]

Petr Hliněný, Filip Pokrývka
2022 arXiv   pre-print
We also prove a similar excluded-subgraphs characterization for hereditary classes of interval graphs with FO model checking in FPT, which concludes the line a research of interval classes with tractable  ...  graph classes, but much less understood on hereditary dense graph classes.  ...  The classes of interval and of interval-overlap graphs have FO interval-like representations.  ... 
arXiv:2204.13742v1 fatcat:sipccgq3mjhhrahthommhl2f5i

Finding a smallest odd hole in a claw-free graph using global structure

Wm. Sean Kennedy, Andrew D. King
2013 Discrete Applied Mathematics  
In this paper we use this result to give an improved shortest-oddhole algorithm for claw-free graphs by exploiting the structural relationship between line graphs and quasi-line graphs suggested by Chudnovsky  ...  and Seymour's structure theorem for quasi-line graphs.  ...  Accordingly, a circular interval graph is a graph with a circular interval representation, i.e.  ... 
doi:10.1016/j.dam.2013.04.026 fatcat:omnzksiuvna2db7kftewnjkhlq

Finding a smallest odd hole in a claw-free graph using global structure [article]

W. Sean Kennedy, Andrew D. King
2011 arXiv   pre-print
In this paper we use this result to give an improved shortest-odd-hole algorithm for claw-free graphs by exploiting the structural relationship between line graphs and quasi-line graphs suggested by Chudnovsky  ...  and Seymour's structure theorem for quasi-line graphs.  ...  Accordingly, a circular interval graph is a graph with a circular interval representation, i.e.  ... 
arXiv:1103.6222v2 fatcat:y4oa7m5f2fhuhkqqyknoiaumsi

Page 45 of Mathematical Reviews Vol. , Issue 92a [page]

1992 Mathematical Reviews  
An interval graph G is homogeneously representable if for every vertex x of G, there exists an interval representation of G with xX corresponding to an end interval.  ...  Tomaz Pisanski (YU-LJUB) 92a:05046 05C10 Olariu, Stephan (1-ODM) On the homogeneous representation of interval graphs. J. Graph Theory 15 (1991), no. 1, 65-80.  ... 

On the recognition of fuzzy circular interval graphs

Gianpaolo Oriolo, Ugo Pietropaoli, Gautier Stauffer
2012 Discrete Mathematics  
Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of clawfree graphs.  ...  In this paper, we provide a polynomial time algorithm for recognizing such graphs, and more importantly for building a suitable model for these graphs.  ...  Therefore, we may associate with a CIG both a representation based on intervals and a representation based on arcs. Proper circular arc graphs have been studied extensively in the last decades.  ... 
doi:10.1016/j.disc.2011.12.029 fatcat:y3af6nzlprbkngbz4okzmmgeya

On the interval number of special graphs

József Balogh, Pascal Ochem, András Pluhár
2004 Journal of Graph Theory  
The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i (G ).  ...  We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independence number.  ...  A representation is depth-two, if every point of the line is covered by at most two intervals. Theorem 1.2 [8] . If a graph G has a depth-two t interval representation, then t ! dðeðGÞ þ 1Þ=vðGÞe.  ... 
doi:10.1002/jgt.20005 fatcat:x5bxdefqujhq3b6zlgoqtkdqwe

Reasoning at the Right Time Granularity [article]

Suchi Saria, Uri Nodelman, Daphne Koller
2012 arXiv   pre-print
Continuous Time Bayesian Networks (CTBNs) avoid time-slicing in the representation by modeling the system as evolving continuously over time.  ...  In this paper, we provide a new EP algorithm that utilizes a general cluster graph architecture where clusters contain distributions that can overlap in both space (set of variables) and time.  ...  Additional partitioning of I jk allows us to have a richer piecewise homogeneous representation over the same interval.  ... 
arXiv:1206.5260v1 fatcat:su2pglen65henhv6rvvge2tlaq
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