When G^2 is a Konig-Egervary graph?

The square of a graph G is the graph G^2 with the same vertex set as in G, and an edge of G^2 is joining two distinct vertices, whenever the distance between them in G is at most 2. G is a square-stable graph if it enjoys the property alpha(G)=alpha(G^2), where alpha(G) is the size of a maximum stable set in G. In this paper we show that G^2 is a Konig-Egervary graph if and only if G is a square-stable Konig-Egervary graph.

💡 Research Summary

The paper investigates the relationship between the square of a graph and the König‑Egerváry property. For a finite, simple, undirected graph G, the square G² is defined on the same vertex set, joining any two distinct vertices whose distance in G is at most two. The stability number α(G) denotes the size of a maximum independent (stable) set, while μ(G) denotes the size of a maximum matching. A graph is König‑Egerváry if α(G)+μ(G)=|V(G)|. The authors call a graph square‑stable when α(G)=α(G²); equivalently, there exists a maximum independent set S in G such that any two distinct vertices of S are at distance at least three (Theorem 1.2).

The first technical result (Lemma 2.1) shows that for any square‑stable graph with at least two vertices, the inequality α(G) ≤ μ(G) holds. The proof constructs, from a maximum independent set S={v₁,…,v_{α(G)}}, pairwise distinct intermediate vertices w_i that lie on shortest paths of length three from each v_i (i<α(G)) to v_{α(G)}. The edges v_i w_i form a matching that saturates S, establishing μ(G)≥α(G).

Proposition 2.3 establishes three equivalent conditions for a graph whose square is König‑Egerváry: (i) α(G)=α(G²), (ii) μ(G)=μ(G²), and (iii) G itself is König‑Egerváry and possesses a perfect matching. The proof uses the basic chain μ(G) ≤ μ(G²) ≤ α(G²) ≤ α(G) together with Lemma 2.1 to derive each implication.

The central theorem (Theorem 2.6) proves that if G² is König‑Egerváry, then G must be a square‑stable König‑Egerváry graph that also has a perfect matching. The argument relies on the known König‑Egerváry decomposition G² = S ∗ H, where S is a maximum independent set of G² and H is the induced subgraph on the remaining vertices. Two claims are shown: (1) each vertex of H is adjacent in G to at most one vertex of S, and (2) the sets of S‑vertices adjacent to H in G and in G² coincide. Consequently, each h∈V(H) is joined by exactly one edge of G to a unique s(h)∈S, and the collection M={h s(h)} forms a matching that is simultaneously maximal in G and G². Hence μ(G)=μ(G²). By Proposition 2.3 this forces α(G)=α(G²), i.e., G is square‑stable, and the equality α(G)=μ(G) yields a perfect matching.

Theorem 2.7 gathers four equivalent characterizations for any graph G with at least two vertices: (i) G² is König‑Egerváry, (ii) G is a square‑stable König‑Egerváry graph, (iii) G has a perfect matching consisting solely of pendant edges, and (iv) G is very well‑covered with exactly α(G) pendant vertices. The equivalence of (ii), (iii), and (iv) had been proved earlier by the authors; the new contribution is the link to (i) via Theorem 2.6.

A notable corollary (Corollary 2.8) specializes the result to trees: the square of a tree T is König‑Egerváry if and only if T is well‑covered. Since well‑covered trees are precisely those with a perfect matching formed by pendant edges (and are automatically very well‑covered), this aligns with the previous equivalences.

In the concluding remarks, the authors observe a chain of inequalities involving several graph invariants: α(G²) ≤ θ(G²) ≤ γ(G) ≤ i(G) ≤ α(G) ≤ θ(G), where θ denotes the clique cover number, γ the domination number, and i the size of a minimal maximal independent set. They note that equality throughout the chain occurs precisely when α(G²)=α(G) or θ(G²)=θ(G), suggesting further avenues for research on how other graph operations affect these invariants and lead to König‑Egerváry structures.