Paired threshold graphs

Ravanmehr, Vida; Puleo, Gregory J.; Bolouki, Sadegh; Milenković, Olgica

doi:10.1016/j.dam.2018.05.008

Cited by 5 publications

(6 citation statements)

References 32 publications

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“…As a matter of fact, all unit interval graphs are paired threshold graphs. This has been observed in [16], and we include a proof because the construction used in it will be exemplary in this paper. An interval model can be specified by the 2n endpoints for the n intervals: The interval for vertex v is denoted by [lp (v), rp (v)], where lp (v) and rp(v) are the, respectively, left point and the right point of the interval.…”

Section: Proposition 21 a Graph G Is A Paired Threshold Graph If And ...mentioning

confidence: 78%

“…Yet another closely related graph class is the difference graphs, defined solely by weight differences [8]. Motivated by applications in social and economic interaction modeling, Ravanmehr et al [16] introduced paired threshold graphs, another generalization of threshold graphs. A graph is a paired threshold graph if there exist a positive vertex weight assignment w and two positive thresholds, T + and T − , such that two vertices are adjacent if and only if the sum of their weights are not less than T + and the difference of their weights are not greater than T − .…”

Section: Introductionmentioning

confidence: 99%

“…For this purpose we may focus on paired threshold graphs that are neither threshold graphs nor unit interval graphs. Ravanmehr et al [16] presented a distance decomposition for such a paired threshold graph G: If G is connected, then they are able to decompose V(G) into a set X, which induces a threshold graph, and a sequence of cliques, of each of which the vertices have the same distance to X.…”

Section: Introductionmentioning

confidence: 99%

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Characterization and Linear-time Recognition of Paired Threshold Graphs

Rong¹,

Cao

Wang³

2019

Preprint

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In a paired threshold graph, each vertex has a weight, and two vertices are adjacent if their weight sum is large enough and their weight difference is small enough. It generalizes threshold graphs and unit interval graphs, both very well studied. We present a vertex ordering characterization of this graph class, which enables us to prove that it is a subclass of interval graphs. Further study of clique paths of paired threshold graphs leads to a simple linear-time recognition algorithm for the class.

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Section: Proposition 21 a Graph G Is A Paired Threshold Graph If And ...mentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characterization and Linear-time Recognition of Paired Threshold Graphs

Rong¹,

Cao

Wang³

2019

Preprint

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“…There are many other generalizations of threshold graphs such as bithreshold graphs [8], threshold signed graphs [3], threshold tolerance graphs [17], quasi-threshold graphs (also known as trivially perfect graphs) [27], weakly threshold graphs [1], paired threshold graphs [20], and mock threshold graphs [2]. We omit the definitions of these graph classes and only note that some small graphs show that these classes are incomparable to the class of double-threshold graphs (e.g., 3K 2 and the bull for bithreshold graphs, 3K 2 and the bull for threshold signed graphs, 2K 2 and the bull for threshold tolerance graphs, C 4 and 2K 3 for quasithreshold graphs, 2K 2 and the bull for weakly threshold graphs, C 4 and the bull for paired threshold graphs, K 3 ∪ C 4 and the bull for mock threshold graphs 1 ).…”

Section: Other Generalizations Of Threshold Graphsmentioning

confidence: 99%

Linear-Time Recognition of Double-Threshold Graphs

et al. 2022

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A graph $$G = (V,E)$$ G = ( V , E ) is a double-threshold graph if there exist a vertex-weight function $$w :V \rightarrow \mathbb {R}$$ w : V → R and two real numbers $$\mathtt {lb}, \mathtt {ub}\in \mathbb {R}$$ lb , ub ∈ R such that $$uv \in E$$ u v ∈ E if and only if $$\mathtt {lb}\le \mathtt {w}(u) + \mathtt {w}(v) \le \mathtt {ub}$$ lb ≤ w ( u ) + w ( v ) ≤ ub . In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in $$O(n^{3} m)$$ O ( n 3 m ) time, where n and m are the numbers of vertices and edges, respectively.

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“…There are many other generalizations of threshold graphs such as threshold signed graphs [2], threshold tolerance graphs [16], quasi-threshold graphs (also known as trivially perfect graphs) [25], weakly threshold graphs [1], and paired threshold graphs [18]. We omit the definitions of these graph classes and only note that some small graphs show that these classes are incomparable to the class of double-threshold graphs (e.g., 3K 2 and bull for threshold signed graphs, 2K 2 and bull for threshold tolerance graphs, C 4 and 2K 3 for quasithreshold graphs, 2K 2 and bull for weakly threshold graphs, C 4 and bull for paired threshold graphs 2 ).…”

Section: Other Generalizations Of Threshold Graphsmentioning

confidence: 99%

Linear-Time Recognition of Double-Threshold Graphs

Kobayashi

Okamoto

Otachi³

et al. 2019

Preprint

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A graph G = (V, E) is a double-threshold graph if there exist a vertex-weight function w : V → R and two real numbers lb, ub ∈ R such that uv ∈ E if and only if lb ≤ w(u) + w(v) ≤ ub. In the literature, those graphs are studied as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs, which gives connections to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [COCOON 2018] ran in O(n 6 ) time, where n is the number of vertices.

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Paired threshold graphs

Cited by 5 publications

References 32 publications

Characterization and Linear-time Recognition of Paired Threshold Graphs

Characterization and Linear-time Recognition of Paired Threshold Graphs

Linear-Time Recognition of Double-Threshold Graphs

Linear-Time Recognition of Double-Threshold Graphs

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