P_4-Colorings and P_4-Bipartite Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.272 ◽

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.

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

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2090 ◽

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.

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Detection number of bipartite graphs and cubic graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.642 ◽

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.

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New Results on Generalized Graph Coloring

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.311 ◽

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.

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Reconstructing a Graph from its Neighborhood Lists

Combinatorics Probability Computing ◽

10.1017/s0963548300000535 ◽

1993 ◽

Vol 2 (2) ◽

pp. 103-113 ◽

Cited By ~ 10

Author(s):

Martin Aigner ◽

Eberhard Triesch

Keyword(s):

Decision Problem ◽

Graph Isomorphism ◽

Bipartite Graphs ◽

Labeled Graph ◽

Np Complete

Associate to a finite labeled graph G(V, E) its multiset of neighborhoods (G) = {N(υ): υ ∈ V}. We discuss the question of when a list is realizable by a graph, and to what extent G is determined by (G). The main results are: the decision problem is NP-complete; for bipartite graphs the decision problem is polynomially equivalent to Graph Isomorphism; forests G are determined up to isomorphism by (G); and if G is connected bipartite and (H) = (G), then H is completely described.

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k-Efficient domination: Algorithmic perspective

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830922500513 ◽

2022 ◽

Author(s):

Mohsen Alambardar Meybodi

Keyword(s):

Decision Problem ◽

Dominating Set ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Chordal Graphs ◽

Sparse Graphs ◽

Fpt Algorithm ◽

Domination Problem ◽

Efficient Dominating Set ◽

Np Complete

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

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A note on compact and compact circular edge-colorings of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.431 ◽

2008 ◽

Vol Vol. 10 no. 3 (Graph and Algorithms) ◽

Author(s):

Dariusz Dereniowski ◽

Adam Nadolski

Keyword(s):

Approximation Algorithm ◽

Decision Problem ◽

Edge Coloring ◽

Exact Algorithm ◽

Weighted Graphs ◽

Edge Colorings ◽

Odd Cycle ◽

International Audience ◽

Np Complete ◽

Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

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On paths, trails and closed trails in edge-colored graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.586 ◽

2012 ◽

Vol Vol. 14 no. 2 (Graph Theory) ◽

Author(s):

Laurent Gourvès ◽

Adria Lyra ◽

Carlos A. Martinhon ◽

Jérôme Monnot

Keyword(s):

Polynomial Time ◽

Optimal Solution ◽

Colored Graph ◽

Edge Disjoint ◽

Colored Graphs ◽

International Audience ◽

Np Complete ◽

T Path ◽

Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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Consecutive Colouring of Oriented Graphs

Results in Mathematics ◽

10.1007/s00025-021-01505-3 ◽

2021 ◽

Vol 76 (4) ◽

Author(s):

Marta Borowiecka-Olszewska ◽

Ewa Drgas-Burchardt ◽

Nahid Yelene Javier-Nol ◽

Rita Zuazua

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Bipartite Graphs ◽

Complete Graphs ◽

Oriented Graphs ◽

Complete Problem ◽

Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

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Pebble Game Algorithms and (k,l)-Sparse Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3394 ◽

2005 ◽

Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽

Author(s):

Audrey Lee ◽

Ileana Streinu

Keyword(s):

Efficient Algorithms ◽

Sparse Graphs ◽

Edge Disjoint ◽

International Audience ◽

Tree Decompositions ◽

Pebble Game

International audience A multi-graph $G$ on n vertices is $(k,l)$-sparse if every subset of $n'≤n$ vertices spans at most $kn'-l$ edges, $0 ≤l < 2k$. $G$ is tight if, in addition, it has exactly $kn - l$ edges. We characterize $(k,l)$-sparse graphs via a family of simple, elegant and efficient algorithms called the $(k,l)$-pebble games. As applications, we use the pebble games for computing components (maximal tight subgraphs) in sparse graphs, to obtain inductive (Henneberg) constructions, and, when $l=k$, edge-disjoint tree decompositions.

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