Connected Counterexamples to the Henning--Yeo Conjecture on Identifying Vertex Covers

Abstract: Henning and Yeo conjectured an upper bound on the identifying vertex cover number of a graph in terms of its order, size, and maximum degree. We disprove the conjectured inequality with a two-parameter family $H_{t,r}$ of connected diameter-two graphs. After clearing denominators, the right-hand side minus the left-hand side is exactly $-(t-1)(r-1)$; hence a connected counterexample exists for every maximum degree at least four. Chaining copies through low-degree vertices preserves the maximum degree and allows the packing number to be determined exactly. At maximum degree five, this gives counterexamples of arbitrarily large order with additive gap $1/13$. For every fixed maximum degree $Δ\ge6$, suitable chains have unbounded additive violation. Thus neither rounding nor a fixed additive correction repairs the conjecture. The supremal normalized additive gap at maximum degree $Δ$ is $Θ(1/Δ)$. An exhaustive check of all graphs of order at most seven shows that the eight-vertex example $H_{2,2}$ has minimum possible order.