On Models of Directed Path Graphs Non Rooted Directed Path Graphs

Gutiérrez, Marisa; Tondato, Silvia Beatriz

doi:10.1007/s00373-015-1600-z

Cited by 2 publications

(3 citation statements)

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“…Proof For Item I, the graph in Figure 4A was presented in [21] as an example of a directed path graph that is not rooted directed path. It turns out that the same graph is also an example of directed path graph that is not in–out rooted directed path.…”

Section: Preliminariesmentioning

confidence: 99%

“…It turns out that the same graph is also an example of directed path graph that is not in–out rooted directed path. This can be seen from the fact that the tree model presented in the figure is the unique model for the graph using directed paths (see [21]), and that it is not an in–out branching. For Item II, Figure 3B presents a tree model of the 4‐sun where the tree is an in–out branching and each path is a directed path in such a tree; in other words, the 4‐sun is an in–out rooted directed path graph.…”

Section: Preliminariesmentioning

confidence: 99%

“…It turns out that the same graph is also an example of directed path graph that is not in-out rooted directed path. This can be seen from the fact that the tree model presented in the figure is the unique model for the graph using directed paths (see [21]), and that it is not an in-out branching.…”

Section: Proposition 2 the Following Relations Hold Between The Menti...mentioning

confidence: 99%

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Parameterized algorithms for Steiner tree and (connected) dominating set on path graphs

de Figueiredo,

Lopes,

de Melo

et al. 2024

Networks

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Chordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Proposition 2 the Following Relations Hold Between The Menti...mentioning

confidence: 99%

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