jaem

TWMS Journal of Applied and Engineering Mathematics

2146-1147 2587-1013

Işık University Press

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Kombinatorik ve Ayrık Matematik (Fiziksel Kombinatorik Hariç)

$S_{4}$ SEPARATION AND P-PARTITION IN ALL-PATH AND DETOUR CONVEXITIES

https://orcid.org/0009-0001-1064-2512

Haponenko

Vladyslav

National University of Kyiv-Mohyla Academy & Kyiv School of Economics

04 07 2026

16 4 443 456 03 06 2025 07 30 2025

2010

TWMS Journal of Applied and Engineering Mathematics

In this work, we consider problems of $S_{4}$ and $p$-convex partition separations with respect to the all-path and the detour convexities. We give characterizations of $p$-all-path convex and $p$-detour convex graphs. With respect to all-path convexity $S_{2}$, $S_{3}$, and $S_{4}$ separable graphs are characterized. Also, we present necessary and sufficient conditions for two sets to be $S_{4}$ separable, for both convexities. Moreover, we prove that in all-path convexity the time complexity of those problems is linear, and it is NP-hard for detour convexity. Finally, we give an algorithm for determining whether two sets in graph are $S_{4}$ separable with respect to all-path convexity.

all-path convexity graph convexity detour convexity convex separation $p$-partition

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