Splitting a graph into disjoint induced paths or cycles

Le, Hoàng‐Oanh; Lê, Van Bang; Müller, Haiko

doi:10.1016/s0166-218x(02)00425-0

Cited by 23 publications

(16 citation statements)

References 8 publications

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“…There are many papers dealing with vertex-partition problems on (di)graphs. Examples (from a long list) are [1,4,5,7,8,9,10,11,12,13,14,15,16,18,20,21,22,23]. Important examples for undirected graphs are bipartite graphs (those having has a 2-partition into two independent sets) and split graphs (those having a 2-partition into a clique and an independent set) [8].…”

Section: Introductionmentioning

confidence: 99%

Finding good 2-partitions of digraphs I. Hereditary properties

Bang‐Jensen

Havet

2016

Theoretical Computer Science

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We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected, connected, minimum outdegree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper, we determine the complexity of of deciding, for any fixed pair of positive integers k1, k2, whether a given digraph has a vertex partition into two digraphs D1, D2 such that |V (Di)| ≥ ki and Di has property Pi for i = 1, 2 when P1 ∈ H and P2 ∈ H ∪ E. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the problems when both P1 and P2 are in E is determined in the companion paper [2].

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Section: Introductionmentioning

confidence: 99%

Finding good 2-partitions of digraphs I. Hereditary properties

Bang‐Jensen

Havet

2016

Theoretical Computer Science

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“…For example, prime number 23 divides the difference of ranks 0 and 161. But, if we can keep only the points (0, x) ∈ W 0,9 and the points (0, y) ∈ W 6,9 such that x and y are not congruent modulo 23, then 23 does not divide gcd (|a 1 − b 1 |, |a 2 − b 2 |) for any a ∈ W 0,9 and b ∈ W 6,9 .…”

Section: Figure 1: Some Grid Drawingsmentioning

confidence: 99%

“…Although the following lemma is already known to be true [4], the known proof is based on the result with so called one-defective colorings. For completeness we include a short proof which uses a similar idea as the previous one (a variation of a technique used by Hoòng-Oanh Le [9]).…”

Section: Figure 4: Construction Of a Planar Graph Which Is Not (1 2)mentioning

confidence: 99%

Grid representations and the chromatic number

Balko

2013

Computational Geometry

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A grid drawing of a graph maps vertices to grid points and edges to line segments that avoid grid points representing other vertices. We show that there is a number of grid points that some line segment of an arbitrary grid drawing must intersect. This number is closely connected to the chromatic number. Second, we study how many columns we need to draw a graph in the grid, introducing some new NP-complete problems. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures asked by

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“…Deciding whether a graph G is a planar graph of cliques is NP-hard, both in this and in Brandenburg's model, and it is NP-hard deciding whether a graph is a path (of any length k ≥ 2) or a tree of paths [13], [37], [39], [48]. However, these results consider infinite types of Y -graphs, such as all cliques or all paths.…”

Section: (Planar K-clique)-clusteringmentioning

confidence: 99%

Visual analysis of large graphs using (X,Y)-clustering and hybrid visualizations

Batagelj¹,

Didimo²,

Liotta³

et al. 2010

2010 IEEE Pacific Visualization Symposium (PacificVis)

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Abstract-Many different approaches have been proposed for the challenging problem of visually analyzing large networks. Clustering is one of the most promising. In this paper we propose a new clustering technique whose goal is that of producing both intra-cluster graphs and inter-cluster graph with desired topological properties. We formalize this concept in the (X,Y )-clustering framework, where Y is the class that defines the desired topological properties of intracluster graphs and X is the class that defines the desired topological properties of the inter-cluster graph. By exploiting this approach hybrid visualization tools can effectively combine different node-link and matrix-based representations, allowing users to interactively explore the graph by expansion/contraction of clusters without loosing their mental map. As a proof of concept, we describe the system VHYXY (Visual Hybrid (X,Y)-clustering) that implements our approach and we present the results of case studies to the visual analysis of social networks.

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Splitting a graph into disjoint induced paths or cycles

Cited by 23 publications

References 8 publications

Finding good 2-partitions of digraphs I. Hereditary properties

Finding good 2-partitions of digraphs I. Hereditary properties

Grid representations and the chromatic number

Visual analysis of large graphs using (X,Y)-clustering and hybrid visualizations

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