[Paper Review] Cops, Robber and Medianwidth Parameters

Konstantinos S. Stavropoulos|arXiv (Cornell University)|Mar 22, 2016

Advanced Graph Theory Research19 references3 citations

TL;DR

This paper introduces $i$-latticewidth, a new graph parameter defined via median decompositions into median graphs isometrically embeddable into the Cartesian product of $i$ paths, forming a hierarchy from pathwidth to clique number. It characterizes $i$-latticewidth as the maximum intersection size of $i$ path decompositions and establishes a direct link to a generalized Cops and Robber game with $i$ cooperating teams of invisible cops, where monotone winning strategies correspond exactly to bounded $i$-latticewidth.

ABSTRACT

In previous work, we introduced median decompositions, a generalisation of tree decompositions where a graph can be modelled after any median graph, along with a hierarchy of $i$-medianwidth parameters $(mw_i)_{i\geq 1}$ starting from treewidth and converging to the clique number. We introduce another graph parameter based on the concept of median decompositions, to be called $i$-latticewidth and denoted by $lw_i$, for which we restrict the modelling median graph of a decomposition to be isometrically embeddable into the Cartesian product of $i$ paths. The sequence $(lw_i)_{i\geq 1}$ gives rise to a hierarchy of parameters starting from pathwidth and converging to the clique number. We characterise the $i$-latticewidth of a graph in terms of maximal intersections of bags of $i$ path decompositions of the graph. We study a generalisation of the classical Cops and Robber game, where the robber plays against not just one, but $i$ cop players. Depending on whether the robber is visible or not, we show a direct connection to $i$-medianwidth or $i$-latticewidth, respectively.

Motivation & Objective

• To define and study a new width parameter, $i$-latticewidth, based on median decompositions into median graphs embeddable into the Cartesian product of $i$ paths.
• To establish a hierarchy of parameters starting from pathwidth and converging to the clique number, analogous to the $i$-medianwidth hierarchy but based on paths instead of trees.
• To characterize $i$-latticewidth in terms of the maximum intersection of $i$ path decompositions of a graph.
• To generalize the Cops and Robber game to $i$ cooperating teams of invisible cops and show that monotone winning strategies in this game characterize $i$-latticewidth.

Proposed method

• Define $i$-latticewidth $\operatorname{lw}_i(G)$ as the minimum $k$ such that there exists a median decomposition of $G$ into a median graph isometrically embeddable into the Cartesian product of $i$ paths, with all bags of size at most $k$.
• Prove that $\operatorname{lw}_i(G)$ equals the maximum size of the intersection $\bigcap_{j=1}^i Z_u^j$ over all $i$-tuples of vertices $u_j$ in $i$ path decompositions of $G$.
• Introduce a generalized Cops and Robber game with $i$ cooperating teams of invisible cops, where each team follows a path decomposition strategy.
• Show that $i$ teams of invisible cops can monotonely search a graph with cooperation at most $k$ if and only if $\operatorname{lw}_i(G) \leq k$, using a strategy adaptation from the classical game.
• Use the concept of convexity and geodesic properties in median graphs to ensure that the decomposition and search strategies preserve structural consistency.
• Leverage the isometric embedding of median graphs into products of paths to relate the game dynamics to the structural parameters of the graph.

Experimental results

Research questions

• RQ1 How can the concept of median decompositions be restricted to median graphs embeddable into products of $i$ paths to define a new width parameter?
• RQ2 What is the structural characterization of $i$-latticewidth in terms of multiple path decompositions of a graph?
• RQ3 Is there a game-theoretic characterization of $i$-latticewidth using a generalized Cops and Robber game with $i$ cooperating teams of invisible cops?
• RQ4 Does the equivalence between monotone and non-monotone winning strategies, known for $i=1$, extend to $i>1$ in the generalized game?
• RQ5 Can obstructing structures or submodular functions be identified for $i$-latticewidth to mirror the role of brambles or connectivity functions in treewidth and pathwidth?

Key findings

• The $i$-latticewidth parameter $\operatorname{lw}_i(G)$ forms a non-increasing hierarchy starting from pathwidth ($\operatorname{pw}(G)+1$) and converging to the clique number $\omega(G)$.
• For any $i < i'$, there exist graph classes of bounded $i'$-latticewidth but unbounded $i$-latticewidth, showing the hierarchy has strongly distinguished levels.
• $\operatorname{lw}_i(G)$ is exactly characterized as the maximum size of the intersection $\bigcap_{j=1}^i Z_u^j$ over all $i$-tuples of vertices $u_j$ in $i$ path decompositions of $G$.
• A graph $G$ can be monotonely searched with cooperation at most $k$ by $i$ teams of invisible cops if and only if $\operatorname{lw}_i(G) \leq k$, establishing a game-theoretic characterization.
• The equivalence between monotone and non-monotone strategies in the classical Cops and Robber game does not necessarily extend to $i>1$, as obstructing notions and submodularity for $i$-latticewidth remain unknown.