Vertex Intersection Graphs of Paths on a Grid: Characterization Within Block Graphs

Alcón, Liliana; Bonomo, Flavia; Mazzoleni, María Pía

doi:10.1007/s00373-017-1791-6

Cited by 13 publications

(7 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A polynomial time decision algorithm for chordal B 0 -VPG graphs is developed in [5]. Characterizations for B 0 -VPG are known within the classes of split graphs, chordal bull-free graphs, chordal claw-free graphs [20] and block graphs [1].…”

Section: Theorem 2 Any Posetmentioning

confidence: 99%

See 1 more Smart Citation

Characterization and a 2D Visualization of B$_0$-VPG Cocomparability Graphs

Pallathumadam¹,

Rajendraprasad²

2020

Preprint

View full text Add to dashboard Cite

B0-VPG graphs are intersection graphs of vertical and horizontal line segments on a plane. Cohen, Golumbic, Trotter, and Wang [Order, 2016] pose the question of characterizing B0-VPG permutation graphs. We respond here by characterizing B0-VPG cocomparability graphs. This characterization also leads to a polynomial time recognition and B0-VPG drawing algorithm for the class. Our B0-VPG drawing algorithm starts by fixing any one of the many posets P whose cocomparability graph is the input graph G. The drawing we obtain not only visualizes G in that one can distinguish comparable pairs from incomparable ones, but one can also identify which among a comparable pair is larger in P from this visualization.

show abstract

Section: Theorem 2 Any Posetmentioning

confidence: 99%

“…A. 1 Step 1. Reduced dd-minor R G of G In this step, we contract a subset of edges of G to obtain a bipartite minor R G of G which satisfies the additional properties listed in Lemma 4.…”

Section: Appendix a Proof Of Correctness Of Algorithmmentioning

confidence: 99%

Characterization and a 2D Visualization of B$_0$-VPG Cocomparability Graphs

Pallathumadam¹,

Rajendraprasad²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…90 degrees turns at a grid-point) that the paths may have: a graph is a B k -VPG graph, for some integer k ≥ 0, if one can assign a path on a grid having at most k bends to each vertex such that two vertices are adjacent if and only if the corresponding paths intersect on at least one grid-point. Since their introduction, B k -VPG graphs have received much attention (see for instance [3,4,[8][9][10][11]17,19,21,22,24]).…”

Section: Introductionmentioning

confidence: 99%

Characterising circular-arc contact B0–VPG graphs

Bonomo

Galby

Gonzalez

2020

Discrete Applied Mathematics

View full text Add to dashboard Cite

A contact B 0 -VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-toone correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown in [15] that Recognition is NP-complete for contact B 0 -VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact B 0 -VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs.

show abstract

“…Since their introduction, EPG and VPG graphs have been extensively studied (see, e.g., [2,3,4,5,6,12,16,17,18,23,24,33]). One of the main considered problems consists in determining the bend-number with respect to EPG (resp.…”

Section: Introductionmentioning

confidence: 99%

CPG graphs: Some structural and hardness results

Champseix¹,

Galby²,

Munaro³

et al. 2019

Preprint

View full text Add to dashboard Cite

In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in [14]. A CPG graph is a graph for which there exists a collection of pairwise interiorly disjoint paths on a grid in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. If every such path has at most k bends for some k ≥ 0, the graph is said to be B k -CPG.We first show that, for any k ≥ 0, the class of B k -CPG graphs is strictly contained in the class of B k+1 -CPG graphs even within the class of planar graphs, thus implying that there exists no k ≥ 0 such that every planar CPG graph is B k -CPG. The main result of the paper is that recognizing CPG graphs and B k -CPG graphs with k ≥ 1 is NP-complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k ≥ 3. We then consider several graph problems restricted to CPG graphs and show, in particular, that Independent Set and Clique Cover remain NP-hard for B 0 -CPG graphs. Finally, we consider the related classes B k -EPG of edge-intersection graphs of paths with at most k bends on a grid. Although it is possible to optimally color a B 0 -EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is NP-complete for B 1 -EPG graphs.

show abstract

Vertex Intersection Graphs of Paths on a Grid: Characterization Within Block Graphs

Cited by 13 publications

References 8 publications

Characterization and a 2D Visualization of B$_0$-VPG Cocomparability Graphs

Characterization and a 2D Visualization of B$_0$-VPG Cocomparability Graphs

Characterising circular-arc contact B0–VPG graphs

CPG graphs: Some structural and hardness results

Contact Info

Product

Resources

About