Discrete-time treatment number

TL;DR

This work introduces the discrete-time treatment model on graphs with three vertex states and defines the treatment number $\\tau_{r,s}(H)$ as the minimal per-step width required to clear a graph under fixed protective length $r$ and vulnerability length $s$. It proves an upper bound $\\tau_{r,s}(H) \\\le\\ \\lceil \\\frac{1+pw(H)}{r+s} \\ ceil$ via path decompositions and shows this bound is tight for cautious protocols; it then analyzes the $r=s=1$ case, giving a complete characterization of graphs with $\\tau(H)=1$ (caterpillars) and computing values for cycles, the subdivided $K'_{1,3}$, and the Petersen graph. It also demonstrates that subdivisions can reduce the treatment number and proves that every tree has a subdivision with $\\tau\le 2$, with constructive methods extending to binary and $m$-ary trees. These results connect graph searching, pathwidth, and isoperimetric bounds to inform efficient containment strategies on networks.

Abstract

We introduce the discrete-time treatment number of a graph, in which each vertex is in exactly one of three states at any given time-step: compromised, vulnerable, or treated. Our treatment number is distinct from other graph searching parameters that use only two states, such as the firefighter problem or Bernshteyn and Lee's inspection number. Vertices represent individuals and edges exist between individuals with close connections. Each vertex starts out as compromised; it can become compromised again even after treatment. Our objective is to treat the entire population so that at the last time-step, no members are vulnerable or compromised, while minimizing the maximum number of treatments that occur at each time-step. This minimum is the treatment number, and it depends on the choice of a pre-determined length of time $r$ that a vertex can remain in a treated state and length of time $s$ that a vertex can remain in a vulnerable state without being treated again. We denote the pathwidth of graph $H$ by $pw(H)$ and prove that the treatment number of $H$ is bounded above by $\lceil \frac{1+pw(H)}{r+s}\rceil$. This equals the best possible lower bound for a cautious treatment plan, defined as one in which each vertex, after being treated for the first time, is treated again within every consecutive $r+s$ time-steps until its last treatment. However, many graphs admit a plan that is not cautious. When $r=s=1$, we find a useful tool for proving lower bounds, show that the treatment number of an $n\times n$ grid equals $\lceil\frac{1+n}{2}\rceil$, characterize graphs that require only one treatment per time-step, and prove that subdividing one edge can reduce the treatment number. It is known that there are trees with arbitrarily large pathwidth; surprisingly, we prove that for any tree $T$, there is a subdivision of $T$ that requires at most two treatments per time-step.

Figures (6)

• Figure 1: A visualization of a $(1,1)$-protocol for $K_{1,3}$, with green vertices also indicated by dots and the yellow vertex with diagonal lines.
• Figure 2: A digraph showing the transitions between the states of treated, vulnerable and compromised.
• Figure 3: A protocol $J$ that clears $H$ and the restricted protocol $J'$ that clears subgraph $H'$. A vertex has label $t$ if it is treated at time-step $t$.
• Figure 4: A minimum width path decomposition of $P_4 \square P_4$. The bold vertices are replacements as we move from left to right in the path.
• Figure 5: The $4\times 4$ grid $P_4 \square P_4$ with one edge subdivided (indicated by a wavy line).

Theorems & Definitions (56)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Definition 2.5
• Definition 2.7
• Proposition 2.8
• proof
• Theorem 2.9
• proof