Intersection Graphs of Non-crossing Paths

Chaplick, Steven

doi:10.1007/978-3-030-30786-8_24

Cited by 4 publications

(1 citation statement)

References 38 publications

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“…Given a collection C of sets, the graph G = (C, {{X, Y } : X, Y ∈ C and X ∩ Y ̸ = ∅}) is called the intersection graph of C. Structural properties and recognition algorithms are known for intersection graphs of (directed) paths in (rooted) trees [9,32,34]. Depending on the collection C, we say that a graph is…”

Section: Preliminariesmentioning

confidence: 99%

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Papadopoulos

Tzimas

2022

Lecture Notes in Computer Science

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Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example that behaves computationally different in subclasses of chordal graph is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph G = (V, E) and a set S ⊆ V , we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains NP-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider variants of the leafage that measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage ℓ, we provide an algorithm for SFVS with running time n O(ℓ) , thus improving upon n O(ℓ 2 ) , the running time of the previously known algorithm obtained for graphs with bounded mim-width. We complement our result by showing that SFVS is W[1]-hard parameterized by ℓ. Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains NP-complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs of mim-width one, which is faster than the previously known algorithm obtained for graphs with bounded mim-width.

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Section: Preliminariesmentioning

confidence: 99%

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Papadopoulos

Tzimas

2022

Lecture Notes in Computer Science

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Intersection Graphs of Non-crossing Paths

Chaplick

2019

Graph-Theoretic Concepts in Computer Science

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We study graph classes modeled by families of non-crossing (NC) connected sets. Two classic graph classes in this context are disk graphs and proper interval graphs. We focus on the cases when the sets are paths and the host is a tree. Forbidden induced subgraph characterizations and linear time certifying recognition algorithms are given for intersection graphs of NC paths of a tree (and related subclasses). For intersection graphs of NC paths of a tree, the dominating set problem is shown to be solvable in linear time. Also, each such graph is shown to have a Hamiltonian cycle if and only if it is 2-connected, and to have a Hamiltonian path if and only if its block-cutpoint tree is a path.

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Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Papadopoulos

Tzimas

2023

Algorithmica

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Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example of a problem that does not behave computationally the same in all subclasses of chordal graphs is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph $$G=(V,E)$$ G = ( V , E ) and a set $$S\subseteq V$$ S ⊆ V , we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains np-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider the leafage, which measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage $$\ell $$ ℓ , we provide an algorithm for solving SFVS with running time $$n^{O(\ell )}$$ n O ( ℓ ) , thus improving upon $$n^{O(\ell ^2)}$$ n O ( ℓ 2 ) , which is the running time of an approach that utilizes the previously known algorithm for graphs with bounded mim-width. We complement our result by showing that SFVS is w[1]-hard parameterized by $$\ell $$ ℓ . Pushing further our positive result, it is natural to also consider the vertex leafage, which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains np-complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for solving SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs with mim-width one, which is faster than the approach of constructing a graph decomposition of mim-width one and applying the previously known algorithm for graphs with bounded mim-width.

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Intersection Graphs of Non-crossing Paths

Cited by 4 publications

References 38 publications

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Intersection Graphs of Non-crossing Paths

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

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