Optimal accessing and non-accessing structures for graph protocols

An accessing set in a graph is a subset B of vertices such that there exists D subset of B, such that each vertex of V\B has an even number of neighbors in D. In this paper, we introduce new bounds on the minimal size kappa’(G) of an accessing set, and on the maximal size kappa(G) of a non-accessing set of a graph G. We show strong connections with perfect codes and give explicitly kappa(G) and kappa’(G) for several families of graphs. Finally, we show that the corresponding decision problems are NP-Complete.

💡 Research Summary

The paper investigates three graph‑theoretic parameters—κ(G), κ′(G) and κ_Q(G)—that arise naturally in graph‑state based quantum secret sharing (QSS) protocols. An “accessing set” B⊆V(G) is defined as a set for which there exists a subset D⊆B with odd cardinality (|D|≡1 mod 2) such that the odd‑neighbourhood Odd(D) is contained in B. Lemma 1 shows that the complement of an accessing set is a non‑accessing set, establishing a duality that underpins the definitions: κ′(G) is the minimum size of an accessing set, κ(G) the maximum size of a non‑accessing set, and κ_Q(G)=max{κ(G), n−κ′(G)} (n=|V(G)|) captures the threshold for QSS protocols.

The authors first study how these parameters behave under graph replication. For G_r, the disjoint union of r copies of a graph G, they prove κ(G_r)=r·κ(G) and κ′(G_r)=κ′(G). This linear scaling is crucial for later NP‑completeness reductions.

Next, they compute κ and κ′ exactly for complete multipartite graphs G_{p,q}, where each of the q parts has size p (so n=pq). The results depend on the parity of q:

• If q is odd, κ(G_{p,q})=n−p and κ′(G_{p,q})=q.
• If q is even, κ(G_{p,q})=max(n−p, n−q) and κ′(G_{p,q})=p+q+1. These formulas reveal a deep connection with perfect codes: a perfect code is an independent set C such that every vertex outside C has exactly one neighbour in C. The authors show that κ(G) reaches its theoretical upper bound n·Δ/(Δ+1) (Δ = maximum degree) if and only if G contains a perfect code in which every code vertex has degree Δ. For regular graphs this condition simplifies to “κ(G)=n·Δ/(Δ+1) ⇔ G has a perfect code”.

Using Lemma 5 they establish the universal inequality κ(G)+κ′(G)≥n. They also derive a general upper bound κ(G)≤n·Δ/(Δ+1) and a lower bound κ′(G)≥n·n/(n−δ) (δ = minimum degree). For regular graphs the lower bound is tight exactly when the complement graph possesses a perfect code.

The computational complexity of the associated decision problems is then addressed. Three problems are defined:

1. KAPPA≥: given (G,k), decide whether κ(G)≥k.
2. KAPPA′≤: given (G,k), decide whether κ′(G)≤k.
3. QKAPPA: given (G,k), decide whether κ_Q(G)≤k (equivalently κ(G)≤k and κ′(G)≥n−k).

Theorem 3 proves KAPPA≥ is NP‑Complete by reduction from the perfect‑code problem on 3‑regular graphs (κ(G)≥3n/4 iff the graph has a perfect code). Theorem 5 shows KAPPA′≤ is NP‑Complete using a similar reduction, with a gadget (adding a K₄) to handle parity issues. Finally, Theorem 6 establishes QKAPPA as co‑NP‑Complete, since a NO‑certificate is either a non‑accessing set of size k−1 or an accessing set of size n−k+1, and hardness follows from the previous reductions.

In summary, the paper introduces novel extremal graph parameters motivated by quantum secret sharing, provides exact values for several graph families, links these parameters to the classical notion of perfect codes, and proves that determining them is computationally intractable (NP‑Complete or co‑NP‑Complete). These results not only deepen the theoretical understanding of graph‑state based QSS but also guide the design of protocols with desirable threshold properties, highlighting the interplay between quantum information theory and combinatorial graph theory.