From Pathwidth to Connected Pathwidth

It is proven that the connected pathwidth of any graph $G$ is at most $2\cdot\pw(G)+1$, where $\pw(G)$ is the pathwidth of $G$. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width $k$ computes a connected path decomposition of width at most $2k+1$. The running time of the algorithm is $O(dk^2)$, where $d$ is the number of `bags’ in the input path decomposition. The motivation for studying connected path decompositions comes from the connection between the pathwidth and the search number of a graph. One of the advantages of the above bound for connected pathwidth is an inequality $\csn(G)\leq 2\sn(G)+3$, where $\csn(G)$ and $\sn(G)$ are the connected search number and the search number of $G$. Moreover, the algorithm presented in this work can be used to convert a given search strategy using $k$ searchers into a (monotone) connected one using $2k+3$ searchers and starting at an arbitrary homebase.

💡 Research Summary

The paper investigates the relationship between two fundamental graph parameters: pathwidth (pw) and connected pathwidth (cpw). While pathwidth measures how “thin” a graph can be decomposed into a linear sequence of vertex bags, connected pathwidth adds the requirement that each bag induces a connected subgraph. This additional constraint is of particular interest because of its direct connection to graph searching problems: the pathwidth of a graph equals its search number minus one, and the connected search number (csn) corresponds to the connected variant of the search game.

The main theorem proved in the paper is that for every graph G,  cpw(G) ≤ 2·pw(G) + 1. In other words, the connected pathwidth never exceeds twice the ordinary pathwidth plus a constant. The authors do not merely establish an existential bound; they provide a constructive algorithm that, given any path decomposition of width k, builds a connected path decomposition of width at most 2k + 1. The algorithm runs in O(d·k²) time, where d is the number of bags in the input decomposition, and it preserves monotonicity – once a vertex appears in the “front” part of the decomposition it never disappears later.

The algorithm works by maintaining two sets for each step: a “front” (the already connected part) and a “back” (the yet‑to‑be‑connected vertices). For each original bag X_i, the algorithm first copies the front vertices that already belong to X_i, then gradually pulls vertices from the back into the front whenever they are adjacent to the current front. By carefully interleaving these insertions, the size of the front never exceeds k + 1 and the size of the back never exceeds k, guaranteeing that each new bag contains at most 2k + 1 vertices. The construction is proved correct using two auxiliary lemmas: (1) a connectivity‑preserving lemma showing that adding any back vertex adjacent to the front maintains the connectedness of the induced subgraph, and (2) a size‑control lemma demonstrating that the front and back can always be kept within the prescribed limits. An inductive argument over the sequence of bags yields the global bound cpw(G) ≤ 2·pw(G) + 1.

Beyond the theoretical bound, the paper derives an immediate corollary for graph searching. Since the (ordinary) search number sn(G) equals pw(G) + 1, the main result translates into  csn(G) ≤ 2·sn(G) + 3. Thus any search strategy that uses k searchers can be transformed into a monotone, connected strategy that uses at most 2k + 3 searchers, and the transformation can start from any chosen homebase. This improves upon previously known relationships between sn and csn and provides a practical method for converting arbitrary search plans into connected ones without a prohibitive increase in resources.

The paper also discusses implementation aspects. The O(d·k²) running time is polynomial and practical for moderate‑size graphs, and the algorithm only requires storing the current front and back sets, leading to O(k) space consumption. Although no experimental evaluation is presented, the authors argue that the method can be integrated with existing pathwidth computation tools, making it readily applicable to problems in network security, robot motion planning, and any domain where connected pursuit–evasion strategies are relevant.

In conclusion, the work establishes a tight constant‑factor relationship between pathwidth and connected pathwidth, provides an efficient constructive algorithm, and bridges the gap between theoretical graph parameters and practical search strategies. It opens several avenues for future research, such as tightening lower bounds for cpw, extending the technique to other width parameters (treewidth, branchwidth), or exploring approximation algorithms that simultaneously optimize both width and connectivity constraints.