Maximum Weight Clique Algorithms for Circular-Arc Graphs and Circle Graphs

1985 ◽  
Vol 14 (1) ◽  
pp. 224-231 ◽  
Author(s):  
Wen-Lian Hsu
Keyword(s):  
Maximum Weight ◽  
Circle Graphs ◽  
Circular Arc ◽  
Circular Arc Graphs ◽  
Maximum Weight Clique

scholarly journals Structural results on circular-arc graphs and circle graphs: A survey and the main open problems

2014 ◽  
Vol 164 ◽  
pp. 427-443 ◽  
Author(s):  
Guillermo Durán ◽  
Luciano N. Grippo ◽  
Martín D. Safe
Keyword(s):  
Open Problems ◽  
Circle Graphs ◽  
Circular Arc ◽  
Circular Arc Graphs

DYNAMIC PROGRAMMING ON INTERVALS

1993 ◽  
Vol 03 (03) ◽  
pp. 323-330 ◽  
Author(s):  
TAKAO ASANO
Keyword(s):  
Dynamic Programming ◽  
Dominating Set ◽  
Minimum Weight ◽  
Time Algorithm ◽  
Interval Graph ◽  
Efficient Implementation ◽  
Maximum Weight ◽  
Circular Arc ◽  
Maximum Weight Clique

We consider problems on intervals which can be solved by dynamic programming. Specifically, we give an efficient implementation of dynamic programming on intervals. As an application, an optimal sequential partition of a graph G=(V, E) can be obtained in O(m log n) time, where n=|V| and m=|E|. We also present an O(n log n) time algorithm for finding a minimum weight dominating set of an interval graph G=(V, E), and an O(m log n) time algorithm for finding a maximum weight clique of a circular-arc graph G=(V, E), provided their intersection models of n intervals (arcs) are given.

Finding a Maximum Clique in a Set of Proper Circular Arcs in Time O(n) with Applications

1997 ◽  
Vol 08 (04) ◽  
pp. 443-467 ◽  
Author(s):  
Glenn K. Manacher ◽  
Terrance A. Mankus
Keyword(s):  
Time Complexity ◽  
Maximum Clique ◽  
Special Property ◽  
Maximum Weight ◽  
Time And Space ◽  
Circular Arc ◽  
Interval Model ◽  
Circular Arcs ◽  
The One ◽  
Circular Arc Graphs

A maximum clique is sought in a set of n proper circular arcs (PCAS). By means of several passes, each O(n) in time and space, a PCAS is transformed initially into a set of circle chords and finally into a set of intervals. This interval model inherits a special property from the PCAS which ensures the discovery of a maximum overlap clique in time O(n). The one-to-one arc/interval correspondence guarantees the identification of the maximum clique in the PCAS in O(n) time and space. The present paper gives new, simpler proofs for the lemmas first outlined by us in Ref. [9], extending the methods outlined in that paper so that the time bound is improved from O(n log n) to O(n). The method depends only on certain interconnections between constructions related to the computation of longest increasing subsequences. Independently, Hell, Huang and Bhattacharya5 recently discovered a completely different approach that also achieves the same complexity, and can moreover be applied to the weighted case and to the coloring problem on proper circular arcs. The previous best result, due to Apostolico and Hambrusch2 applies to general circular arc models and has time complexity O(n2 log log n) and space complexity O(n). As applications of the method, we show that maximum weight clique of a set of weighted proper circular arcs can be found in time O(n2) and space O(n). The previous best result was O(n2 log log n) for dense general circular arc graphs.13 We also show that, for n chords with randomly placed endpoints (1) the average cardinality of a maximum clique is cn1/2 ± o(n1/2), where 21/2< c < e21/2, and (2) a maximum clique may be found in average time O(n3/2) and space θ(n). The previous best average time complexity, derived from Ref. [1], was O(n3/2 log n).

scholarly journals Graphs of low chordality

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Sunil Chandran ◽  
Vadim V. Lozin ◽  
C.R. Subramanian
Keyword(s):  
Chordal Graphs ◽  
Circle Graphs ◽  
Circular Arc ◽  
International Audience ◽  
Circular Arc Graphs

International audience The chordality of a graph with at least one cycle is the length of the longest induced cycle in it. The odd (even) chordality is defined to be the length of the longest induced odd (even) cycle in it. Chordal graphs have chordality at most 3. We show that co-circular-arc graphs and co-circle graphs have even chordality at most 4. We also identify few other classes of graphs having bounded (by a constant) chordality values.

scholarly journals A Complementary Pivoting Approach to the Maximum Weight Clique Problem

2002 ◽  
Vol 12 (4) ◽  
pp. 928-948 ◽  
Author(s):  
Alessio Massaro ◽  
Marcello Pelillo ◽  
Immanuel M. Bomze
Keyword(s):  
Maximum Weight ◽  
Complementary Pivoting ◽  
Maximum Weight Clique ◽  
Clique Problem

Surjective L (2, 1)-labeling of cycles and circular-arc graphs

2018 ◽  
Vol 35 (1) ◽  
pp. 739-748 ◽  
Author(s):  
Sk. Amanathulla ◽  
Madhumangal Pal
Keyword(s):  
Circular Arc ◽  
Circular Arc Graphs
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