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The one-arm event $A_R$ is the event that a specified root is joined through open vertices or edges to graph distance $R$.

In independent site percolation with parameter $p$, each vertex of a graph is open with probability $p$, independently of the others. Retain those vertices and every original edge whose two endpoints are retained. This gives an induced subgraph, whose connected components of a graph are the percolation clusters. A closed vertex is absent from the usable network, whereas bond percolation keeps all vertices and instead removes individual edges.

A specified simple graph path of $m$ edges has $m+1$ distinct vertices. Its probability of being entirely open is therefore $p^{m+1}$, including the starting vertex. Conditional on that vertex being open, it is $p^m$ by independence. For a two-edge graph path at $p=1/2$, the unconditional probability is $1/8$, compared with $1/4$ in bond percolation. This comparison concerns a fixed route; percolation critical probabilities require analysis of all possible routes on the specified infinite graph.

In independent bond percolation with parameter $p$, every edge of a graph is open with probability $p$ and closed with probability $1-p$. The choices are independent random variables with a Bernoulli distribution. All vertices remain present; only the usable connections change. The percolation clusters are the connected components of a graph formed by the open edges, including isolated vertices.

A specified simple graph path with $m$ edges is an open path with probability $p^m$, by independence. Connecting its endpoints is a different event: other graph paths may also connect them. The four-edge example in percolation theory makes that distinction explicit. Interpreting open edges as usable channels makes the model useful for connectivity questions; it does not by itself assign a travel time, flow rate, or capacity to a channel.

At unit spacing, exactness through cubic polynomials gives

An $n$-vertex graph containing no path of length $t$ has at most $(t-1)n/2$ edges.

This path-length bound is a different result from the Erdős-Gallai theorem on degree sequences.

Every connected $n$-vertex graph of minimum degree $\delta$ contains a path of length at least $\min (2\delta ,n-1)$.

The cycle graph $C_n$ has vertices $v_1,\dots ,v_n$ and edges $v_i{v}_{i+1}$, with indices read cyclically.