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

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.

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

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.

scholarly journals Pebble game algorithms and sparse graphs

2008 ◽  
Vol 308 (8) ◽  
pp. 1425-1437 ◽  
Author(s):  
Audrey Lee ◽  
Ileana Streinu
Keyword(s):  
Sparse Graphs ◽  
Pebble Game

scholarly journals Edge Disjoint Hamilton Cycles in Knödel Graphs

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Palanivel Subramania Nadar Paulraja ◽  
S Sampath Kumar
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Cycle Decomposition ◽  
Edge Disjoint ◽  
International Audience ◽  
Appl Math

International audience The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

scholarly journals Binary Labelings for Plane Quadrangulations and their Relatives

2011 ◽  
Vol Vol. 12 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Stefan Felsner ◽  
Clemens Huemer ◽  
Sarah Kappes ◽  
David Orden
Keyword(s):  
Bipartite Graphs ◽  
Plane Graphs ◽  
International Audience ◽  
Tree Decompositions

Graphs and Algorithms International audience Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder's one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2-book. Furthermore, as Schnyder labelings have been extended to 3-connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2-connected bipartite graphs.

The linear 2-arboricity of sparse graphs

2017 ◽  
Vol 09 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Yuanchao Li ◽  
Xiaoxue Hu
Keyword(s):  
Sparse Graphs ◽  
Edge Disjoint ◽  
Linear Formula

The linear [Formula: see text]-arboricity [Formula: see text] of a graph [Formula: see text] is the least integer [Formula: see text] such that [Formula: see text] can be partitioned into [Formula: see text] edge-disjoint forests, whose components are paths of length at most 2. In this paper, we study the linear [Formula: see text]-arboricity of sparse graphs, and prove the following results: (1) let [Formula: see text] be a 2-degenerate graph, we have [Formula: see text]; (2) if [Formula: see text], then [Formula: see text]; (3) if [Formula: see text], then [Formula: see text]; (4) if [Formula: see text], then [Formula: see text]; (5) if [Formula: see text], then [Formula: see text].

scholarly journals On P_4-tidy graphs

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
V. Giakoumakis ◽  
F. Roussel ◽  
H. Thuillier
Keyword(s):  
Optimization Problems ◽  
Linear Time ◽  
Hamiltonian Path ◽  
Clique Number ◽  
Stability Number ◽  
Sparse Graphs ◽  
Modular Decomposition ◽  
New Class ◽  
International Audience ◽  
Linear Algorithms

International audience We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].

scholarly journals Multicolored isomorphic spanning trees in complete graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Gregory Constantine
Keyword(s):  
Complete Graph ◽  
Spanning Trees ◽  
Complete Graphs ◽  
Edge Disjoint ◽  
International Audience

International audience Can a complete graph on an even number n (>4) of vertices be properly edge-colored with n-1 colors in such a way that the edges can be partitioned into edge disjoint colorful isomorphic spanning trees? A spanning treee is colorful if all n-1 colors occur among its edges. It is proved that this is possible to accomplish whenever n is a power of two, or five times a power of two.

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