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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.  ...

### On Homogeneously Representable Interval Graphs

Marcin Krzywkowski, Jerzy Topp
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.  ...

### 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  ...

### 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.  ...

### 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] ).  ...

### 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  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...
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