
Subset Feedback Vertex Set on Graphs of Bounded Independent Set Size
The (Weighted) Subset Feedback Vertex Set problem is a generalization of...
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Reconfiguration of Colorable Sets in Classes of Perfect Graphs
A set of vertices in a graph is ccolorable if the subgraph induced by t...
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Maximizing Communication Throughput in Tree Networks
A widely studied problem in communication networks is that of finding th...
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On Some Combinatorial Problems in Cographs
The family of graphs that can be constructed from isolated vertices by d...
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LengthBounded Cuts: Proper Interval Graphs and Structural Parameters
In the presented paper we study the LengthBounded Cut problem for speci...
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Flip distances between graph orientations
Flip graphs are a ubiquitous class of graphs, which encode relations ind...
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Graph Similarity and Homomorphism Densities
We introduce the tree distance, a new distance measure on graphs. The tr...
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Computing Subset Feedback Vertex Set via Leafage
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 wellknown 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 graph G=(V,E) and a set S⊆ V, SFVS asks for a minimum set of vertices that intersects all cycles containing a vertex of S. SFVS is known to be polynomialtime solvable on interval graphs, whereas SFVS remains 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). Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the smallest number among the maximum number of leaves of all subtrees in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains complete on undirected path graphs, i.e., graphs having vertex leafage at most two. Moreover, we strengthen previouslyknown polynomialtime algorithm for SFVS on directed path graphs that form a proper subclass of undirected path graphs and graphs of mimwidth one.
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