Pré-Publication, Document De Travail Année : 2025

On the star forest polytope for 4-cactus graphs

Résumé

This paper investigates the polyhedral structure of the Maximum Weight Star Forest Problem (MWSFP) in an undirected weighted graph G = (V, E), where each edge has a non-negative weight. A star in G is either an isolated node or a connected subgraph in which all edges share a common endpoint, and a star forest is a collection of disjoint stars. The objective of the MWSFP is to find a star forest with the maximum total edge weight. This problem is NP-hard in general but can be solved in polynomial time when G is a cactus graph [19]. In this paper, we provide a complete polyhedral description of the star forest polytope SF P (G) when G is a 4-cactus graph, a subclass of cactus graphs where each cycle has at most four edges. More precisely, we introduce a new class of facet-defining inequalities, called M -cactus inequalities, which hold for any graph. We then show that when G is a 4cactus graph, the M -cactus inequalities, together with the non-negativity inequalities, completely describe SF P (G).

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hal-05284746 , version 1 (26-09-2025)

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  • HAL Id : hal-05284746 , version 1

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Thanh Loan Nguyen, Viet Hung Nguyen, Minh Hieu Nguyen, Thi Viet Thanh Vu. On the star forest polytope for 4-cactus graphs. 2025. ⟨hal-05284746⟩
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