Elusive properties of countably infinite graphs
Márton Elekes, Tamás Kátay, Anett Kocsis
TL;DR
The paper extends the Aanderaa–Karp–Rosenberg framework to countably infinite graphs by studying the infinite-length Seeker–Hider game on $V=\mathbb{N}$ and introducing $\infty$-elusiveness. It proves a spectrum of results: (i) natural properties like containing cycles (girth), connectedness, and bipartiteness exhibit $\infty$-elusiveness; (ii) there exist monotone properties not $\infty$-elusive, including the presence of $k$ independent edges and absence of isolated vertices; (iii) a separation result showing $\mathcal{S}_0$ is $\infty$-elusive but not elusive, thereby distinguishing $\infty$-elusiveness from traditional elusiveness; (iv) a descriptive-set-theoretic analysis clarifies when $\infty$-elusive games are determined, linking determinacy to monotonicity and Borel complexity, with open problems about the full scope of determinacy and the existence of natural $\mathcal{S}$ exhibiting these phenomena. Overall, the work reveals a nuanced landscape where infinite-horizon strategies produce richer behavior than their transfinite or finite analogs, with implications for understanding graph properties under adversarial exploration and for connections to descriptive set theory. The results broaden understanding of graph properties in the infinite setting and highlight key open questions at the intersection of combinatorics, logic, and topology.
Abstract
A graph property is elusive (or evasive) if any algorithm testing it by asking questions of the form ''Is there an edge between vertices x and y?'' must, in the worst case, examine all pairs of vertices. Elusiveness for infinite vertex sets has been first studied by Csernák and Soukup, who proved that the long-standing Aanderaa-Karp-Rosenberg Conjecture -- which states that every nontrivial monotone graph property is elusive -- fails for infinite vertex sets. We extend their work by giving a closer look to the case when the vertex set is countably infinite and the ''algorithm'' terminates after infinitely many steps. Among others, we prove that connectedness is elusive, which strengthens a result of Csernák and Soukup. We give counterexamples to the infinite version of the Aanderaa-Karp-Rosenberg Conjecture even if the ''algorithm'' is required to terminate after infinitely many steps, which strengthens results of Csernák and Soukup.