--- title: Leaky Forcing in Graphs url: https://www.emergentmind.com/topics/leaky-forcing type: topic --- # Leaky Forcing in Graphs Searching arXiv for recent papers on leaky forcing and closely related variants. Leaky forcing is a fault-tolerant variant of zero forcing on a graph in which an initial blue set must color the entire graph even after an adversary designates up to $\ell$ vertices as leaks, meaning that those vertices do not participate in forcing. The associated parameter, the $\ell$-leaky forcing number $Z_{(\ell)}(G)$, extends the zero forcing number by $Z_{(0)}(G)=Z(G)$ and satisfies the monotone chain $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$. Introduced to model unreliable junctures in networks, leaky forcing has developed into a framework for robustness in combinatorial propagation, with recursive characterizations, fort-based duality, equivalence among several leak models, connections to strong structural controllability, and exact or sharp results for multiple graph families [1910.00168, 2008.06552, 2009.07073, 2303.02366]. ## 1. Core definitions and forcing mechanism In the zero forcing game on a simple graph $G=(V,E)$, one begins with an initial blue set $B\subseteq V$. At each step, a blue vertex $v$ can force a neighbor $u$ if $u$ is its only uncolored neighbor. A set that eventually colors every vertex blue is a zero forcing set, and the minimum cardinality of such a set is the zero forcing number $Z(G)$. Leaky forcing modifies this rule by allowing up to $\ell$ leaks to be placed after the initial blue vertices are chosen. A leaky vertex does not participate in forcing; one formulation describes vertices with leaks as behaving as if they are attached to an extra neighbor, making the color change rule stricter. A set $B_\ell$ is an $\ell$-leaky forcing set if, regardless of which $\ell$ vertices become leaky, the process still colors all vertices, and $Z_{(\ell)}(G)$ denotes the minimum size of such a set [2202.12671]. The same framework is also written in terms of a fixed leak set $L\subset V$: the leaky derived set $\mathcal{D}_L(V')$ is the set of vertices eventually colored by zero forcing when vertices in $L$ are not allowed to force. In this notation, $V'$ is an $\ell$-leaky forcing set precisely when $\mathcal{D}_L(V')=V$ for every $L$ of size $\ell$. This formulation is especially useful when leaky forcing is imported into network-control problems, where the initial blue set is interpreted as a leader set or input set [2303.02366]. ## 2. Characterizations, forts, and resilience A central structural fact is that leaky forcing is governed by redundancy of possible parents in forcing processes. If $B$ is an $\ell$-leaky forcing set, then for every vertex $v\in V(G)\setminus B$, there must be at least $\ell+1$ distinct blue vertices that can force $v$ in possible forcing processes. For $\ell=1$, this becomes an exact characterization: $B$ is a $1$-leaky forcing set if and only if every $v\notin B$ can be forced in at least two ways, by two different blue vertices in some possible processes. More generally, $B$ is an $\ell$-leaky forcing set if and only if it is an $(\ell-1)$-leaky forcing set and, for every set $L$ of $\ell-1$ leaks and every $v\notin B$, there exist $xv,yv\in \mathcal{F}_L(B)$ with $x\neq y$ [2008.06552, 2009.07073]. The theory also admits a fort formulation. In the leaky setting, forts are subsets that can block propagation under adversarial leak placement, and a set is an $\ell$-leaky forcing set if and only if it intersects every $\ell$-leaky fort. This dual viewpoint underlies both exact proofs and optimization formulations. Closely related is the degree obstruction: any $\ell$-leaky forcing set must contain all vertices of degree at most $\ell$, and if $\ell\geq \Delta(G)$ then $Z_{(\ell)}(G)=|V(G)|$ [1910.00168, 2508.02564]. Resilience is the special case in which the fault-tolerant parameter does not increase relative to ordinary zero forcing. A graph $G$ is called $\ell$-resilient if \[ Z_{(\ell)}(G)=Z_{(0)}(G). \] This condition imposes immediate structural restrictions. In particular, if $G$ is $\ell$-resilient, then $\delta(G)\geq \ell+1$ [2008.06552]. A plausible implication is that resilience is best viewed not as a marginal property of a specific forcing chronology but as a global redundancy condition on the set of all admissible chronologies. ## 3. Equivalent leak models and controllability interpretations One of the most important generalizations is that the location of the fault can be modeled in several formally different ways without changing the minimum size of a robust forcing set. Besides vertex leaks, an edge leak is an edge $xy$ across which neither $x\to y$ nor $y\to x$ is allowed, and a specified leak is a single prohibited direction $vu$. The resulting parameters are the $\ell$-edge-leaky forcing number $Z'_{(\ell)}(G)$, the specified $\ell$-leaky forcing number $Z^s_{(\ell)}(G)$, and the mixed $\ell$-leaky forcing number $Z^m_{(\ell)}(G)$ when leak types are combined. The main equivalence theorem states that for every graph $G$ and $\ell\geq 0$, \[ Z_{(\ell)}(G)=Z'_{(\ell)}(G)=Z^s_{(\ell)}(G)=Z^m_{(\ell)}(G). \] This shows that the worst-case number of disruptions, rather than their type, controls the parameter [2009.07073]. An analogous equivalence appears in strong structural controllability. For a graph $G=(V,E)$, with graph-constrained symmetric matrices $\mathcal{M}(G)$ and dynamics \[ \dot{x}(t)=Mx(t)+Bu(t), \qquad M\in \mathcal{M}(G), \] a leader set $V'\subset V$ renders the network strong structurally controllable if and only if $V'$ is a zero forcing set. In the resilient setting, three threat models are considered: leaky nodes, non-forcing edges, and removable edges. The main theorem states that for a fixed $V'$ and $\ell$, the following are equivalent: $V'$ is an $\ell$-LFS, $V'$ is an $\ell$-EFS, and $V'$ is an $\ell$-FSR. Thus resilience to $\ell$ misbehaving nodes, $\ell$ non-forcing edges, and $\ell$ edge removals coincide at the level of leader selection [2303.02366]. ## 4. Exact values and bounds for major graph families For several classical graph families, the leaky forcing number is known exactly. For paths, \[ Z_{(\ell)}(P_n)= \begin{cases} 1, & \ell=0\\ 2, & \ell=1\\ n, & \ell\geq 2, \end{cases} \] and for cycles, \[ Z_{(\ell)}(C_n)= \begin{cases} 2, & \ell=0,1\\ n, & \ell\geq 2. \end{cases} \] For complete graphs, \[ Z_{(\ell)}(K_n)= \begin{cases} n-1, & \ell\leq n-2\\ n, & \ell\geq n-1, \end{cases} \] while for wheel graphs, \[ Z_{(\ell)}(W_n)= \begin{cases} 3, & \ell=0,1\\ \lceil 2n/3\rceil+1, & \ell=2\\ n, & n>\ell>2\\ n+1, & \ell=n,n+1. \end{cases} \] For trees, if $T$ is a tree and $B$ is the set of vertices of degree at most $\ell$, then $Z_{(\ell)}(T)=|B|$; in particular, for a tree with $t$ leaves, excluding $P_2$, one has $Z_{(1)}(T)=t$, and the optimal forcing sets are exactly the leaves [1910.00168, 2008.06552]. For two-dimensional products and related families, the emphasis has been on one-leak robustness. For all $1\leq n\leq m$, \[ Z_{(1)}(P_n \square P_m)\leq \min\{2n,m\}, \] resolving the question of whether a grid could exceed that upper bound; if $m\geq 2n^2$, then $Z_{(1)}(P_n \square P_m)=2n$. For supertriangles $T_n$, the known values and bounds are \[ Z_{(0)}(T_n)=Z_{(1)}(T_n)=n,\qquad Z_{(2)}(T_n)\leq Z_{(3)}(T_n)\leq 2n-1,\qquad Z_{(4)}(T_n)=Z_{(5)}(T_n)=3n-3. \] These results are typical of the subject: explicit constructions are paired with two-direction forcing arguments that certify robustness to leaks [2008.06552]. The hypercube has been a recurrent test case. For the $d$-dimensional hypercube $Q_d$, it is proved that \[ Z_{(d-2)}(Q_d)\leq 2^{d-1}, \] by coloring one $(d-1)$-dimensional subcube and then using a combinatorial counting argument on the remaining vertices. Earlier computations showed this bound is tight for $Q_3$, $Q_4$, and $Q_5$, and the conjecture is \[ Z_{(d-2)}(Q_d)=2^{d-1}\qquad \text{for all } d\geq 2. \] The same paper observes that the result automatically yields upper bounds for $\ell