On the Computational Complexity of the Strong Geodetic Recognition Problem

Carlos V. G. C. Lima; Vinicius F. dos Santos; João H. G. Sousa; Sebastián A. Urrutia

Home/Archive/arXiv

arXiv · 2208.01796

On the Computational Complexity of the Strong Geodetic Recognition Problem

Carlos V. G. C. Lima·Vinicius F. dos Santos·João H. G. Sousa·Sebastián A. Urrutia

View original source ↗Read source document ↗Export metadata ↓

Abstract

A strong geodetic set of a graph~$G=(V,E)$ is a vertex set~$S \subseteq V(G)$ in which it is possible to cover all the remaining vertices of~$V(G) \setminus S$ by assigning a unique shortest path between each vertex pair of~$S$. In the Strong Geodetic problem (SG) a graph~$G$ and a positive integer~$k$ are given as input and one has to decide whether~$G$ has a strong geodetic set of cardinality at most~$k$. This problem is known to be NP-hard for general graphs. In this work we introduce the Strong Geodetic Recognition problem (SGR), which consists in determining whether even a given vertex set~$S \subseteq V(G)$ is strong geodetic. We demonstrate that this version is NP-complete. We investigate and compare the computational complexity of both decision problems restricted to some graph classes, deriving polynomial-time algorithms, NP-completeness proofs, and initial parameterized complexity results, including an answer to an open question in the literature for the complexity of SG for chordal graphs.

Explore related subjects

cs.CC

Keep this discovery

BibTeX RIS

Carlos V. G. C. Lima, Vinicius F. dos Santos, João H. G. Sousa, Sebastián A. Urrutia. 2022-08-03. On the Computational Complexity of the Strong Geodetic Recognition Problem. https://arxiv.org/abs/2208.01796

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗