Linear time algorithms for counting the number of minimal vertex covers with minimum/maximum size in an interval graph

2008 ◽  
Vol 107 (6) ◽  
pp. 257-264 ◽  
Author(s):  
Min-Sheng Lin ◽  
Yung-Jui Chen
Keyword(s):  
Linear Time ◽  
Maximum Size ◽  
Interval Graph ◽  
Linear Time Algorithms

scholarly journals Closed graphs are proper interval graphs

2014 ◽  
Vol 22 (3) ◽  
pp. 37-44
Author(s):  
Marilena Crupi ◽  
Giancarlo Rinaldo
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Simple Graph ◽  
Interval Graphs ◽  
Closed Graph ◽  
Graph Recognition ◽  
Proper Interval Graph ◽  
Linear Time Algorithms

Abstract Let G be a connected simple graph. We prove that G is a closed graph if and only if G is a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.

scholarly journals Minimal Obstructions for Partial Representations of Interval Graphs

10.37236/5862 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Pavel Klavik ◽  
Maria Saumell
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Interval Graphs ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Intersection Graphs ◽  
Certifying Algorithm ◽  
Partial Representation ◽  
Partial Representation Extension ◽  
Linear Time Algorithms

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals.  We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension.

scholarly journals Almost Linear Time Algorithms for Minsum k-Sink Problems on Dynamic Flow Path Networks

2021 ◽  
Author(s):  
Yuya Higashikawa ◽  
Naoki Katoh ◽  
Junichi Teruyama ◽  
Koji Watase
Keyword(s):  
Linear Time ◽  
Flow Path ◽  
Dynamic Flow ◽  
Linear Time Algorithms

scholarly journals Linear-Time Algorithms for Tree Root Problems

Algorithmica ◽  
2013 ◽  
Vol 71 (2) ◽  
pp. 471-495 ◽  
Author(s):  
Maw-Shang Chang ◽  
Ming-Tat Ko ◽  
Hsueh-I Lu
Keyword(s):  
Linear Time ◽  
Linear Time Algorithms

FAULT TOLERANT ROUTING IN HYPERCUBES AND STAR GRAPHS

1996 ◽  
Vol 06 (01) ◽  
pp. 127-136 ◽  
Author(s):  
QIAN-PING GU ◽  
SHIETUNG PENG
Keyword(s):  
Free Path ◽  
Fault Tolerant ◽  
Linear Time ◽  
Time Efficiency ◽  
Routing Problem ◽  
Star Graphs ◽  
Linear Time Algorithms ◽  
Better Than

In this paper, we give two linear time algorithms for node-to-node fault tolerant routing problem in n-dimensional hypercubes Hn and star graphs Gn. The first algorithm, given at most n−1 arbitrary fault nodes and two non-fault nodes s and t in Hn, finds a fault-free path s→t of length at most [Formula: see text] in O(n) time, where d(s, t) is the distance between s and t. Our second algorithm, given at most n−2 fault nodes and two non-fault nodes s and t in Gn, finds a fault-free path s→t of length at most d(Gn)+3 in O(n) time, where [Formula: see text] is the diameter of Gn. When the time efficiency of finding the routing path is more important than the length of the path, the algorithms in this paper are better than the previous ones.

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