scholarly journals Clique cycle transversals in graphs with few P₄'s

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Raquel Bravo ◽  
Sulamita Klein ◽  
Loana Tito Nogueira ◽  
Fábio Protti
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Finite Family ◽  
Feedback Vertex Set ◽  
Induced Subgraphs ◽  
Complete Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Vertex Set ◽  
International Audience ◽  
Acyclic Subgraph

Graph Theory International audience A graph is extended P4-laden if each of its induced subgraphs with at most six vertices that contains more than two induced P4's is 2K2,C4-free. A cycle transversal (or feedback vertex set) of a graph G is a subset T ⊆ V (G) such that T ∩ V (C) 6= ∅ for every cycle C of G; if, in addition, T is a clique, then T is a clique cycle transversal (cct). Finding a cct in a graph G is equivalent to partitioning V (G) into subsets C and F such that C induces a complete subgraph and F an acyclic subgraph. This work considers the problem of characterizing extended P4-laden graphs admitting a cct. We characterize such graphs by means of a finite family of forbidden induced subgraphs, and present a linear-time algorithm to recognize them.

scholarly journals A new characterization and a recognition algorithm of Lucas cubes

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Andrej Taranenko
Keyword(s):  
Graph Theory ◽  
Interconnection Networks ◽  
Linear Time ◽  
Recognition Algorithm ◽  
Induced Subgraphs ◽  
Fibonacci Cube ◽  
Vertex Set ◽  
International Audience ◽  
Binary Strings

Graph Theory International audience Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube. This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.

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 Probe split graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Van Bang Le ◽  
H. N. Ridder
Keyword(s):  
Undirected Graph ◽  
Recognition Algorithm ◽  
Not Given ◽  
Induced Subgraphs ◽  
Split Graph ◽  
Forbidden Induced Subgraphs ◽  
Recognition Algorithms ◽  
Vertex Set ◽  
International Audience ◽  
Split Graphs

Graphs and Algorithms International audience An undirected graph G=(V,E) is a probe split graph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and there exists E' ⊆ N× N such that G'=(V,E∪ E') is a split graph. Recently Chang et al. gave an O(V4(V+E)) time recognition algorithm for probe split graphs. In this article we give O(V2+VE) time recognition algorithms and characterisations by forbidden induced subgraphs both for the case when the partition into probes and non-probes is given, and when it is not given.

scholarly journals A linear time algorithm for the minimum Weighted Feedback Vertex Set on diamonds

2005 ◽  
Vol 94 (1) ◽  
pp. 29-35 ◽  
Author(s):  
F. Carrabs ◽  
R. Cerulli ◽  
M. Gentili ◽  
G. Parlato
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Feedback Vertex Set ◽  
Vertex Set

scholarly journals A Linear Time Algorithm for a Variant of the MAX CUT Problem in Series Parallel Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-4 ◽  
Author(s):  
Brahim Chaourar
Keyword(s):  
Linear Time ◽  
Maximum Flow ◽  
Time Algorithm ◽  
Minimum Cut ◽  
Linear Time Algorithm ◽  
Induced Subgraphs ◽  
Positive Weight ◽  
Max Cut Problem ◽  
In Series ◽  
Minimum Cut Problem

Given a graph G=V,E, a connected sides cut U,V\U or δU is the set of edges of E linking all vertices of U to all vertices of V\U such that the induced subgraphs GU and GV\U are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that wΩ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.

scholarly journals Graphs and complete intersection toric ideals

2015 ◽  
Vol 14 (09) ◽  
pp. 1540011 ◽  
Author(s):  
I. Bermejo ◽  
I. García-Marco ◽  
E. Reyes
Keyword(s):  
Complete Intersection ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Toric Ideals ◽  
Induced Subgraphs ◽  
Toric Ideal ◽  
Vertex Set ◽  
Combinatorial Point ◽  
Ring Graph

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal PG is a complete intersection or not. Whenever PG is a complete intersection, the algorithm also returns a minimal set of generators of PG. Moreover, we prove that if G is a connected graph and PG is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals Packing triangles in low degree graphs and indifference graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gordana Manić ◽  
Yoshiko Wakabayashi
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Set Packing ◽  
The Best Approximation ◽  
Low Degree ◽  
International Audience ◽  
Approximation Guarantee ◽  
Vertex Disjoint

International audience We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

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