A tie-break model for graph search

In this paper, we consider the problem of the recognition of various kinds of orderings produced by graph searches. To this aim, we introduce a new framework, the Tie-Breaking Label Search (TBLS), in order to handle a broad variety of searches. This new model is based on partial orders defined on the label set and it unifies the General Label Search (GLS) formalism of Krueger, Simonet and Berry (2011), and the “pattern-conditions” formalism of Corneil and Krueger (2008). It allows us to derive some general properties including new pattern-conditions (yielding memory-efficient certificates) for many usual searches, including BFS, DFS, LBFS and LDFS. Furthermore, the new model allows easy expression of multi-sweep uses of searches that depend on previous (search) orderings of the graph’s vertex set.

💡 Research Summary

The paper introduces a unifying framework for graph search orderings called Tie‑Break Label Search (TBLS). A TBLS instance is defined by a graph G = (V,E), a strict partial order ≺ on the family of finite subsets of positive integers (the label set), and a global tie‑break permutation τ over the vertices. Each vertex’s label at step i is the set of visitation times of its already visited neighbours; the algorithm repeatedly selects the set of “eligible” vertices whose labels are maximal with respect to ≺, and then picks the leftmost vertex of this set according to τ. This makes the search deterministic, eliminating the arbitrary choices present in many classic algorithms.

The authors show that TBLS subsumes the General Label Search (GLS) model of Krueger, Simonet and Berry (2011) and the pattern‑condition formalism of Corneil and Krueger (2008). They prove that the two models generate exactly the same class of searches (Section 5). The key technical tool is Property 3.1, which characterises an S‑ordering (the ordering produced by a TBLS with partial order ≺ =S) by the condition that for any two vertices x and y with x preceding y in the ordering, the label of x (at the moment of its selection) is not strictly smaller than the label of y with respect to ≺. This condition is equivalent to the classic pattern‑conditions (e.g., the 4‑point condition for LBFS) but expressed purely in terms of set‑based labels.

The paper defines an “extension” relation between two TBLS searches: S′ extends S iff every ordering produced by S′ is also an ordering of S. Theorem 3.5 establishes that this extension relation coincides exactly with the inclusion of the underlying partial orders (i.e., ≺{S′} extends ≺{S}). Consequently, the hierarchy of classic searches (BFS ⊂ LBFS ⊂ MNS, etc.) can be understood as a hierarchy of increasingly refined partial orders on label sets.

Concrete instantiations of ≺ are given for several well‑known searches:

• BFS – ≺ is set inclusion; a vertex with a larger neighbourhood (in terms of visited vertices) is preferred.
• DFS – ≺ is reverse inclusion, reflecting the depth‑first preference for vertices whose visited neighbourhood is a subset of another’s.
• LBFS – ≺ is the lexicographic order on the sorted list of visitation times (implemented as a set order).
• LDFS – ≺ is the reverse lexicographic order, mirroring the depth‑first analogue of LBFS. For each case the authors derive new pattern‑conditions that serve as memory‑efficient certificates: the label set itself can be stored instead of a full string of timestamps.

A major contribution is the treatment of multi‑sweep algorithms. By choosing τ as the reverse of the ordering obtained in a previous sweep, TBLS naturally models “+‑searches” (e.g., LBFS⁺, where ties are broken by the rightmost vertex in the previous ordering). This allows the authors to express sequences such as σ₀ arbitrary, σ_{i+1}=TBLS(G, ≺, reverse(σ_i)) and to prove that after a bounded number of sweeps the resulting ordering has desirable structural properties (e.g., unit‑interval ordering after three LBFS sweeps on a unit‑interval graph).

The paper also provides a simple recognition theorem (Theorem 3.7): a permutation σ is a TBLS ordering for a given ≺ iff running TBLS with τ = σ reproduces σ. This self‑referential test eliminates the need for external simulation and yields an efficient way to verify whether a given vertex ordering could arise from a particular search. The authors illustrate its use in proving non‑TBLS orderings for certain graphs.

Implementation aspects are discussed in Section 7. Labels are stored as unordered sets, and the partial order comparison can be accelerated using hash‑based representations of the subsets. The tie‑break permutation τ can be maintained as an array, and the overall algorithm runs in linear time for the classic searches because the partial orders used there admit constant‑time comparisons. The authors also sketch how the framework can be parallelised: each processor can maintain local label sets and synchronize only when determining the global maximal label class.

In summary, the paper offers a clean, mathematically elegant abstraction of graph searches that unifies previous formalisms, clarifies the relationships among classic searches through partial‑order extensions, supports multi‑sweep strategies via a built‑in tie‑break mechanism, and provides practical tools for recognizing and certifying search orderings. The TBLS model has the potential to simplify both theoretical analyses of graph algorithms and their concrete implementations, especially in contexts where deterministic tie‑breaking and memory‑efficient certificates are essential.