scholarly journals Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Oleg Duginov
Keyword(s):  
Bipartite Graphs ◽  
Partition Problem ◽  
Induced Subgraphs ◽  
Permutation Graphs ◽  
Vertex Partition ◽  
Vertex Set ◽  
International Audience ◽  
Bipartite Permutation Graphs ◽  
Np Complete ◽  
Polynomially Solvable

Graph Theory International audience Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.

scholarly journals On bipartite powers of bigraphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Yoshio Okamoto ◽  
Yota Otachi ◽  
Ryuhei Uehara
Keyword(s):  
Graph Theory ◽  
Permutation Graphs ◽  
International Audience ◽  
Bipartite Permutation Graphs ◽  
Np Complete

Graph Theory International audience The notion of graph powers is a well-studied topic in graph theory and its applications. In this paper, we investigate a bipartite analogue of graph powers, which we call bipartite powers of bigraphs. We show that the classes of bipartite permutation graphs and interval bigraphs are closed under taking bipartite power. We also show that the problem of recognizing bipartite powers is NP-complete in general.

scholarly journals P_4-Colorings and P_4-Bipartite Graphs

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Chinh T. Hoàng ◽  
Van Bang Le
Keyword(s):  
Bipartite Graphs ◽  
Perfect Graph ◽  
Class Meeting ◽  
Sparse Graphs ◽  
Color Class ◽  
Vertex Partition ◽  
International Audience ◽  
Np Complete ◽  
Chordless Path

International audience A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.

scholarly journals New Results on Generalized Graph Coloring

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Vladimir E. Alekseev ◽  
Alastair Farrugia ◽  
Vadim V. Lozin
Keyword(s):  
Graph Coloring ◽  
Bipartite Graphs ◽  
Np Hard ◽  
Graph Classes ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete

International audience For graph classes \wp_1,...,\wp_k, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V_1,...,V_k so that V_j induces a graph in the class \wp_j (j=1,2,...,k). If \wp_1=...=\wp_k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k≥ 3. Recently, this result has been generalized by showing that if all \wp_i's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs. Clearly, a similar result follows when all the \wp_i's are co-additive.

scholarly journals Convex Partitions of Graphs induced by Paths of Order Three

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
C. C. Centeno ◽  
S. Dantas ◽  
M. C. Dourado ◽  
Dieter Rautenbach ◽  
Jayme Luiz Szwarcfiter
Keyword(s):  
Convex Sets ◽  
Graph Classes ◽  
Convex Partitions ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete

Graphs and Algorithms International audience A set C of vertices of a graph G is P(3)-convex if v is an element of C for every path uvw in G with u, w is an element of C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P(3)-convex sets. Furthermore, we study such partitions for a variety of graph classes.

scholarly journals Representations of Edge Intersection Graphs of Paths in a Tree

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Charles Golumbic ◽  
Marina Lipshteyn ◽  
Michal Stern
Keyword(s):  
Analogous Result ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Network Applications ◽  
Host Trees ◽  
Complete Problems ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete ◽  
Simple Paths

International audience Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.

scholarly journals A Note on the Speed of Hereditary Graph Properties

10.37236/644 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vadim V. Lozin ◽  
Colin Mayhill ◽  
Victor Zamaraev
Keyword(s):  
Practical Importance ◽  
Bipartite Graphs ◽  
Hereditary Property ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Properties ◽  
Graph Property ◽  
Vertex Set ◽  
Split Graphs ◽  
Hereditary Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

scholarly journals Combinatorial Game Theory Foundations Applied to Digraph Kernels

10.37236/1325 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Aviezri S. Fraenkel
Keyword(s):  
Game Theory ◽  
Opposite Direction ◽  
Decision Problem ◽  
Independent Interest ◽  
Combinatorial Game Theory ◽  
Restricted Class ◽  
Combinatorial Game ◽  
Induced Subgraphs ◽  
Vertex Set ◽  
Np Complete

Known complexity facts: the decision problem of the existence of a kernel in a digraph $G=(V,E)$ is NP-complete; if all of the cycles of $G$ have even length, then $G$ has a kernel; and the question of the number of kernels is $\#$P-complete even for this restricted class of digraphs. In the opposite direction, we construct game theory tools, of independent interest, concerning strategies in the presence of draw positions, to show how to partition $V$, in $O(|E|)$ time, into $3$ subsets $S_1,S_2,S_3$, such that $S_1$ lies in all the kernels; $S_2$ lies in the complements of all the kernels; and on $S_3$ the kernels may be nonunique. Thus, in particular, digraphs with a "large" number of kernels are those in which $S_3$ is "large"; possibly $S_1=S_2=\emptyset$. We also show that $G$ can be decomposed, in $O(|E|)$ time, into two induced subgraphs $G_1$, with vertex-set $S_1\cup S_2$, which has a unique kernel; and $G_2$, with vertex-set $S_3$, such that any kernel $K$ of $G$ is the union of the kernel of $G_1$ and a kernel of $G_2$. In particular, $G$ has no kernel if and only if $G_2$ has none. Our results hold even for some classes of infinite digraphs.

scholarly journals Partitioning a split graph into induced subgraphs isomorphic to the path of order 3.

2019 ◽  
Vol 55 (1) ◽  
pp. 32-49
Author(s):  
O. I. Duginov
Keyword(s):  
Computational Complexity ◽  
Efficient Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Team Formation ◽  
Urgent Problem ◽  
Induced Subgraphs ◽  
Split Graph ◽  
Vertex Set ◽  
Polynomially Solvable

The study of the computational complexity of problems on graphs is an urgent problem. We show that the problem of deciding whether the vertex set of a given split graph of order 3n can be partitioned into induced subgraphs isomorphic to P3 is a polynomially solvable problem. We develop a polynomial-time algorithm based on the method of augmenting graphs. The developed efficient algorithm can be used for solving team formation problems.

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.

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