scholarly journals Linear-Time Recognition of Double-Threshold Graphs

Algorithmica ◽  
2022 ◽  
Author(s):  
Yusuke Kobayashi ◽  
Yoshio Okamoto ◽  
Yota Otachi ◽  
Yushi Uno
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Real Numbers ◽  
Vertex Weight ◽  
Threshold Graphs ◽  
Bipartite Permutation Graphs ◽  
Threshold Graph ◽  
Double Threshold

AbstractA 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.

A LINEAR-TIME ALGORITHM FOR TESTING THE INSCRIBABILITY OF TRIVALENT POLYHEDRA

1995 ◽  
Vol 05 (01n02) ◽  
pp. 21-36 ◽  
Author(s):  
MICHAEL B. DILLENCOURT ◽  
WARREN D. SMITH
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Combinatorial Characterization ◽  
Integer Arithmetic

We present an algorithm for testing the inscribability of a trivalent polyhedron or, equivalently, testing the circumscribability of a simplicial polyhedron. Our algorithm, which runs in linear time and uses only low-precision integer arithmetic, is based on a purely combinatorial characterization of inscribable trivalent polyhedra.

scholarly journals Linear Time Recognition of P4-Indifferent Graphs

1999 ◽  
Vol 6 (38) ◽  
Author(s):  
Romeo Rizzi
Keyword(s):  
Key Words ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Induced Subgraphs ◽  
Modular Decomposition ◽  
Forbidden Induced Subgraphs ◽  
Chordless Path

<p>A simple graph is P4-indifferent if it admits a total order < on<br />its nodes such that every chordless path with nodes a, b, c, d and edges<br />ab, bc, cd has a < b < c < d or a > b > c > d. P4-indifferent graphs generalize<br /> indifferent graphs and are perfectly orderable. Recently, Hoang,<br />Maray and Noy gave a characterization of P4-indifferent graphs in<br />terms of forbidden induced subgraphs. We clarify their proof and describe<br /> a linear time algorithm to recognize P4-indifferent graphs. When<br />the input is a P4-indifferent graph, then the algorithm computes an order < as above.</p><p>Key words: P4-indifference, linear time, recognition, modular decomposition.</p><p> </p>

scholarly journals Proper Orientations of Chordal Graphs

2020 ◽  
Author(s):  
Julio Araujo ◽  
Alexandre Cezar ◽  
Carlos Vinícius Gomes Costa Lima ◽  
Vinicius Fernandes Dos Santos ◽  
Ana Shirley Ferreira Silva
Keyword(s):  
Upper Bound ◽  
Linear Time ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Linear Time Algorithm ◽  
Threshold Graphs ◽  
Minimum Integer ◽  
Split Graphs ◽  
Np Complete ◽  
Proper Orientation

An orientation D of a graph G = (V, E) is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each v ∈ V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u) ≠ dD−(v), for all uv ∈ E(G). An orientation with maximum indegree at most k is called a k-orientation. The proper orientation number of G, denoted by χ→(G), is the minimum integer k such that G admits a proper k-orientation. We prove that determining whether χ→(G) ≤ k is NP-complete for chordal graphs of bounded diameter. We also present a tight upper bound for χ→(G) on split graphs and a linear-time algorithm for quasi-threshold graphs.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

A Linear time algorithm for deciding security

1976 ◽  
Author(s):  
A. K. Jones ◽  
R. J. Lipton ◽  
L. Snyder
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm

AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

2000 ◽  
Vol 11 (03) ◽  
pp. 365-371 ◽  
Author(s):  
LJUBOMIR PERKOVIĆ ◽  
BRUCE REED
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Tree Decomposition ◽  
Linear Time Algorithm ◽  
Input Graph ◽  
Primary Motivation ◽  
Small Width ◽  
Tree Decompositions ◽  
Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

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