scholarly journals An O(nm)-time certifying algorithm for recognizing HHD-free graphs

2012 ◽  
Vol 452 ◽  
pp. 117-131 ◽  
Author(s):  
Stavros D. Nikolopoulos ◽  
Leonidas Palios
Keyword(s):  
Certifying Algorithm

LDFS-Based Certifying Algorithm for the Minimum Path Cover Problem on Cocomparability Graphs

2013 ◽  
Vol 42 (3) ◽  
pp. 792-807 ◽  
Author(s):  
Derek G. Corneil ◽  
Barnaby Dalton ◽  
Michel Habib
Keyword(s):  
Path Cover ◽  
Minimum Path ◽  
Certifying Algorithm ◽  
Cocomparability Graphs ◽  
Cover Problem

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.

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