[Paper Review] New $\mathcal{F}$-Saturation Games on Directed Graphs

Jonathan D. Lee, Ago-Erik Riet|arXiv (Cornell University)|Sep 1, 2014

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper introduces and analyzes new saturation games on directed graphs, focusing on avoiding directed paths $P_k$ and orientation-saturated subgraphs in undirected graphs. It establishes tight bounds for the directed $\mathcal{F}$-homomorphism-saturation game on $K_n$ avoiding $P_k$, and provides asymptotically tight bounds for the biased orientation-saturation game on $K_n$ avoiding $P_{k+1}$, with conjectured optimal behavior matching a refined parameter $\lambda = \frac{2a}{b+2a}$. The results reveal critical dependence on graph structure and game bias, extending classical saturation game theory to directed and oriented settings.

ABSTRACT

We study analogues of $\mathcal{F}$-saturation games, first introduced by Furedi, Reimer and Seress in 1991, and named as such by West. We examine analogous games on directed graphs, and show tight results on the walk-avoiding game. We also examine an intermediate game played on undirected graphs, such that there exists an orientation avoiding a given family of directed graphs, and show bounds on the score. This last game is shown to be equivalent to a recent game studied by Hefetz, Krivelevich, Naor and Stojakovic, and we give new bounds for biased versions of this game.

Motivation & Objective

• To extend classical $\mathcal{F}$-saturation games to directed graphs, introducing the $\mathcal{F}$-homomorphism-saturation game on digraphs.
• To analyze the structure of $P_k$-homomorphism-saturated digraphs and derive exact game scores for $K_n$.
• To introduce and study an orientation-saturation game on undirected graphs, where a subgraph is saturated if some orientation avoids a family of forbidden directed graphs.
• To establish tight asymptotic bounds for the biased orientation-saturation game $\mathcal{G}_o^{a,b}(K_n, P_{k+1})$ under the condition $a,b \ll k \ll n$.
• To conjecture a refined asymptotic formula for the game score that unifies the upper and lower bounds via a parameter $\lambda = \frac{2a}{b+2a}$.

Proposed method

• Characterize $P_k$-homomorphism-saturated digraphs as induced by a total order on vertex classes $G_1, \ldots, G_{k-1}$, where each class corresponds to the length of the longest descending path ending at a vertex.
• Use a vertex partitioning strategy with red and blue edges: red edges within clusters, blue edges between clusters, to control degree ratios under player moves.
• Design a strategy for Prolonger to maintain the constraint $d_{\text{Red}}(v) \geq c \cdot d_{\text{Blue}}(v)$ with $c = \lfloor a/(2b) \rfloor$, ensuring that red edges can be added without violating saturation.
• Design a strategy for Shortener to maintain $d_{\text{Blue}}(v) \geq c' \cdot d_{\text{Red}}(v)$ with $c' = \lfloor b/(2a) \rfloor$, forcing the formation of large color classes and limiting the number of available edges.
• Apply induction and degree counting arguments to bound the number of edges that can be added before saturation, using the cluster size $k\lambda^{-}a - 1$ for Prolonger and $(1 - \lambda^{+})k - b - 1$ for Shortener.
• Derive asymptotic bounds using the constants $\lambda^{-} = \frac{\lfloor b/(2a) \rfloor}{1 + \lfloor b/(2a) \rfloor}$ and $\lambda^{+} = \frac{1}{1 + \lfloor a/(2b) \rfloor}$, leading to bounds of the form $\binom{n}{2}(1 - 1/\lambda^{\pm}k)(1 + o(1))$.

Experimental results

Research questions

• RQ1 What is the score of the directed $\mathcal{F}$-homomorphism-saturation game on $K_n$ when $\mathcal{F} = \{P_k\}$, and how does it depend on $k$ and $n$?
• RQ2 How does the game score of the orientation-saturation game on $K_n$ vary under biased play, where Prolonger takes $a$ moves and Shortener takes $b$ moves consecutively?
• RQ3 Can the bounds for the biased orientation-saturation game be tightened to match a conjectured asymptotic formula involving $\lambda = \frac{2a}{b + 2a}$?
• RQ4 What structural properties characterize $P_k$-homomorphism-saturated digraphs, and how can they be leveraged to design optimal strategies for Prolonger and Shortener?
• RQ5 How does the behavior of the orientation-saturation game compare to the purely directed game, particularly in terms of dependence on $k$?

Key findings

• The score of the directed $\mathcal{F}$-homomorphism-saturation game on $K_n$ avoiding $P_k$ is $0$ for $k \leq 2$, $\lfloor n^2/4 \rfloor$ for $k = 3$, and $\frac{1}{3}n^2 + \frac{1}{3}nk + O(n + k^2)$ for $k \geq 4$, showing a sharp transition at $k=3$.
• For the biased orientation-saturation game $\mathcal{G}_o^{a,b}(K_n, P_{k+1})$ with $a,b \ll k \ll n$, the game score is bounded below by $\binom{n}{2}(1 - 1/\lambda^{-}k)(1 + o(1))$ and above by $\binom{n}{2}(1 - 1/\lambda^{+}k)(1 + o(1))$, where $\lambda^{-} = \frac{\lfloor b/(2a) \rfloor}{1 + \lfloor b/(2a) \rfloor}$ and $\lambda^{+} = \frac{1}{1 + \lfloor a/(2b) \rfloor}$.
• The conjectured optimal score is $\binom{n}{2}(1 - 1/\lambda k)(1 + o(1))$ with $\lambda = \frac{2a}{b + 2a}$, suggesting a refined and symmetric behavior under bias.
• The proof strategies for Prolonger and Shortener rely on maintaining degree ratio constraints across red and blue edge partitions, with the bounds sensitive to the ratio $a/b$, reflecting strategic trade-offs in move efficiency.
• The orientation-saturation game exhibits strong dependence on $k$, unlike the purely directed game, which is nearly independent of $k$, indicating that orientation constraints significantly alter game dynamics.
• The analysis reveals that the primary weakness in the current bounds lies in the inability to handle small violations of degree constraints, suggesting that optimal play may allow for more strategic edge placement that simultaneously corrects multiple imbalances.