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 recognition of generalized Fibonacci cubes $Q_h(111)$

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Yoomi Rho ◽  
Aleksander Vesel
Keyword(s):  
Linear Time ◽  
Binary String ◽  
Recursive Structure ◽  
Fibonacci Cube ◽  
Vertex Set ◽  
International Audience ◽  
Binary Strings

International audience The generalized Fibonacci cube $Q_h(f)$ is the graph obtained from the $h$-cube $Q_h$ by removing all vertices that contain a given binary string $f$ as a substring. In particular, the vertex set of the 3rd order generalized Fibonacci cube $Q_h(111)$ is the set of all binary strings $b_1b_2 \ldots b_h$ containing no three consecutive 1's. We present a new characterization of the 3rd order generalized Fibonacci cubes based on their recursive structure. The characterization is the basis for an algorithm which recognizes these graphs in linear time.

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 On probe 2-clique graphs and probe diamond-free graphs

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Flavia Bonomo ◽  
Celina M. H. Figueiredo ◽  
Guillermo Duran ◽  
Luciano N. Grippo ◽  
Martín D. Safe ◽  
...  
Keyword(s):  
Graph Theory ◽  
Recognition Algorithm ◽  
Chordal Graphs ◽  
Stable Set ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Block Graphs ◽  
International Audience ◽  
Clique Graphs ◽  
Probe Block

Graph Theory International audience Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.

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 p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals Edge stability in secure graph domination

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Anton Pierre Burger ◽  
Alewyn Petrus Villiers ◽  
Jan Harm Vuuren
Keyword(s):  
Graph Theory ◽  
Dominating Set ◽  
Domination Number ◽  
Arbitrary Subset ◽  
Graph Domination ◽  
Vertex Set ◽  
International Audience ◽  
Edge Stability ◽  
Secure Domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

scholarly journals On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Olivier Baudon ◽  
Julien Bensmail ◽  
Rafał Kalinowski ◽  
Antoni Marczyk ◽  
Jakub Przybyło ◽  
...  
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Vertex Set ◽  
International Audience ◽  
Positive Integers

Graph Theory International audience A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ {1,\textellipsis,k}, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.

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 Recognizing HH-free, HHD-free, and Welsh-Powell Opposition Graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Stavros D. Nikolopoulos ◽  
Leonidas Palios
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Recognition Algorithm ◽  
Space Complexity ◽  
Linear Time Algorithm ◽  
Free Graph ◽  
Graph Recognition ◽  
International Audience ◽  
Time And Space Complexity ◽  
Perfect Elimination Ordering

International audience In this paper, we consider the recognition problem on three classes of perfectly orderable graphs, namely, the HH-free, the HHD-free, and the Welsh-Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in O(n min \m α (n,n), m + n^2 log n\) time whether a given graph G on n vertices and m edges contains a house or a hole; this implies an O(n min \m α (n,n), m + n^2 log n\)-time and O(n+m)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement øverlineG of the graph G is HH-free can be efficiently resolved in O(n m) time using O(n^2) space, which leads to an O(n m)-time and O(n^2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n^3) time and O(n^2) space.

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