[Paper Review] Weak and Strong k-connectivity games

Asaf Ferber, Dan Hefetz|arXiv (Cornell University)|Mar 15, 2012

Complexity and Algorithms in Graphs5 references4 citations

TL;DR

This paper establishes that Maker can win the weak $k$-vertex-connectivity game on $K_n$ in at most $\lfloor kn/2\rfloor + 1$ moves, which is optimal, and Red can win the corresponding strong $k$-vertex-connectivity game within the same bound. The strategy leverages a structured, stage-based approach to maintain degree control and ensure $k$-connectivity with minimal moves.

ABSTRACT

For a positive integer $k$ we consider the $k$-vertex-connectivity game, played on the edge set of $K_n$, the complete graph on $n$ vertices. We first study the Maker-Breaker version of this game and prove that, for any integer $k \geq 2$ and sufficiently large $n$, Maker has a strategy for winning this game within $\lfloor k n/2 floor + 1$ moves, which is clearly best possible. This answers a question of Hefetz, Krivelevich, Stojaković and Szabó. We then consider the strong $k$-vertex-connectivity game. For every positive integer $k$ and sufficiently large $n$, we describe an explicit first player's winning strategy for this game.

Motivation & Objective

• To resolve an open question on whether Maker can win the weak $k$-vertex-connectivity game within $kn/2 + c_k$ moves for a constant $c_k$ independent of $n$.
• To provide an explicit, fast winning strategy for the strong $k$-vertex-connectivity game, extending known results from weak games.
• To demonstrate that the bound $\lfloor kn/2\rfloor + 1$ is tight for both weak and strong versions of the $k$-connectivity game.
• To explore the connection between fast winning strategies in weak games and their potential to generate explicit winning strategies in strong games.

Proposed method

• Design a stage-based strategy for Maker in the weak $k$-connectivity game, dividing play into sub-stages focused on degree control and connectivity maintenance.
• Use degree-based control: ensure that in Maker’s graph, no vertex exceeds degree $k$, and only two vertices have degree $k-1$ at critical stages.
• Apply a strategy stealing argument and hypergraph coloring techniques to rule out draws in the strong game, ensuring Red has a winning strategy.
• Leverage Theorem 3.5 (on degree bounds and graph expansion) to guarantee progress in later sub-stages of the strategy.
• Use the fact that $\Delta(B) \leq k+1$ and $|U| = \Omega(n)$ to ensure Red can complete the final stages of the strategy.
• Prove optimality by showing that if the game lasts exactly $kn/2 + 1$ moves, then Breaker/Blue must exceed degree $k$ in their graph, implying Maker/Red wins.

Experimental results

Research questions

• RQ1 Can Maker win the weak $k$-vertex-connectivity game on $K_n$ within $\lfloor kn/2\rfloor + 1$ moves for all $k \geq 2$ and sufficiently large $n$?
• RQ2 Is it possible to construct an explicit winning strategy for Red in the strong $k$-vertex-connectivity game, given a fast strategy in the weak version?
• RQ3 What is the minimal number of moves required for Maker to achieve $k$-vertex-connectivity in the edge set of $K_n$?
• RQ4 Can the dependency on $n$ in the excess moves beyond $kn/2$ be eliminated in the weak $k$-connectivity game?
• RQ5 Does a fast winning strategy in a weak game always imply the existence of an explicit winning strategy in the corresponding strong game?

Key findings

• Maker can win the weak $k$-vertex-connectivity game on $K_n$ within $\lfloor kn/2\rfloor + 1$ moves for all $k \geq 2$ and sufficiently large $n$, which is optimal.
• This bound is tight, as Maker cannot win in fewer than $kn/2$ moves due to the minimum edge requirement for $k$-connectivity.
• Red has an explicit winning strategy in the strong $k$-vertex-connectivity game on $K_n$ within $\lfloor kn/2\rfloor + 1$ moves for $k \geq 3$ and sufficiently large $n$.
• The strategy ensures that at most two vertices in Red’s graph have degree $k-1$ at critical stages, and degree control prevents Blue from blocking progress.
• The proof shows that if the game lasts exactly $kn/2 + 1$ moves, then Breaker/Blue must have a vertex of degree $>k$, implying the game ends in a win for Maker/Red.
• The result extends to the minimum-degree-$k$ game: both Maker and Red can win within $\lfloor kn/2\rfloor + 1$ moves, with a more natural strategy than direct reduction from $k$-connectivity.