On the Terminal Connection Problem

Melo, Alexsander Andrade de; Figueiredo, Celina M. H. de; Souza, Uéverton S.

doi:10.1007/978-3-030-67731-2_20

Cited by 2 publications

(1 citation statement)

References 25 publications

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“…The NP-completeness proof for TCP on chordal bipartite graphs were published in , and all the other mentioned results were published in [de Melo et al 2020, de Melo et al 2021a, de Melo et al 2023]. In particular, in [de Melo et al 2020], we analyse the complexity of S-TCP, the strict variant of TCP.…”

Section: Connection Problemsmentioning

confidence: 99%

On (in)tractability of connection and cut problems

Melo

Figueiredo

Souza³

et al. 2023

Anais Do XXXVI Concurso De Teses E Dissertações (CTD 2023)

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This work addresses connection and cut problems from the viewpoint of graph classes and computational complexity, classic and parameterized. Regarding connection problems, we investigate the so-called TERMINAL CONNECTION problem (TCP), which can be seen as a generalisation of the classical STEINER TREE problem. We propose several complexity results for TCP, when restricted to specific graph classes, and some of its input parameters are fixed. As for cut problems, we analyse the complexity of the classical MAXCUT problem. We introduce the first complexity classification for the problem on interval graphs of bounded interval count. In addition, we prove the NP-completeness of MAXCUT on permutation graphs, settling a question posed by David S. Johnson in the Ongoing Guide to NP-completeness, which has been open since 1985. Finally, we consider the problem of computing the zig-zag number of a directed graph, which is a directed width measure defined over cuts of a graph.

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Section: Connection Problemsmentioning

confidence: 99%

On (in)tractability of connection and cut problems

Melo

Figueiredo

Souza³

et al. 2023

Anais Do XXXVI Concurso De Teses E Dissertações (CTD 2023)

View full text Add to dashboard Cite

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On the computational difficulty of the terminal connection problem

Melo

Figueiredo

Souza

2023

RAIRO-Theor. Inf. Appl.

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A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) ⊆ W ⊆ V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The Terminal connection problem (TCP) asks whether G admits a connection tree for W with at most ℓ linkers and at most r routers, while the Steiner tree problem asks whether G admits a connection tree for W with at most k non-terminal vertices. We prove that, if r ≥ 1 is fixed, then TCP is polynomial-time solvable when restricted to split graphs. This result separates the complexity of TCP from the complexity of Steiner tree, which is known to be NP-complete on split graphs. Additionally, we prove that TCP is NP-complete on strongly chordal graphs, even if r ≥ 0 is fixed, whereas Steiner tree is known to be polynomial-time solvable. We also prove that, when parameterized by clique-width, TCP is W[1]-hard, whereas STeiner tree is known to be in FPT. On the other hand, agreeing with the complexity of Steiner tree, we prove that TCP is linear-time solvable when restricted to cographs (i.e. graphs of clique-width 2). Finally, we prove that, even if either ℓ ≥ 0 or r ≥ 0 is fixed, TCP remains NP-complete on graphs of maximum degree 3.

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On the Terminal Connection Problem

Cited by 2 publications

References 25 publications

On (in)tractability of connection and cut problems

On (in)tractability of connection and cut problems

On the computational difficulty of the terminal connection problem

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