Token sliding independent set reconfiguration on block graphs

TL;DR

The study proves that token sliding independent set reconfiguration is solvable in polynomial time on block graphs by combining a local analysis around cut-vertices with a global potential framework. It introduces depth-based p-independence, capacity, and attack notions, and defines pot(C,p) as the maximal reachable capacity at each p, computable in $O(|E(G)|^4)$ time via Compute-potentials. A rigidity-based pruning step identifies vertices that cannot participate in any reconfiguration sequence, and a subgraph reduction preserves potentials to enable modular, componentwise checks. The algorithm ultimately decides reachability between two independent sets by aligning rigid structures and verifying per-component cardinalities, yielding a definitive Yes/No in $O(|E(G)|^4)$ time. The work generalizes the tree-case to block graphs, marking the first substantial extension beyond trees for this problem, while leaving open questions about guaranteed short reconfiguration sequences in block graphs.

Abstract

Let $S$ be an independent set of a simple undirected graph $G$. Suppose that each vertex of $S$ has a token placed on it. The tokens are allowed to be moved, one at a time, by sliding along the edges of $G$, so that after each move, the vertices having tokens always form an independent set of $G$. We would like to determine whether the tokens can be eventually brought to stay on the vertices of another independent set $S'$ of $G$ in this manner. In other words, we would like to decide if we can transform $S$ into $S'$ through a sequence of steps, each of which involves substituting a vertex in the current independent set with one of its neighbours to obtain another independent set. This problem of determining if one independent set of a graph ``is reachable'' from another independent set of it is known to be PSPACE-hard even for split graphs, planar graphs, and graphs of bounded treewidth. Polynomial time algorithms have been obtained for certain graph classes like trees, interval graphs, claw-free graphs, and bipartite permutation graphs. We present a polynomial time algorithm for the problem on block graphs, which are the graphs in which every maximal 2-connected subgraph is a clique. Our algorithm is the first generalization of the known polynomial time algorithm for trees to a larger class of graphs (note that trees form a proper subclass of block graphs).

Figures (5)

• Figure 1: Diagram showing tokens placed in a part of a block graph (tokens represented by black squares)
• Figure 2: The token on $x$ is not trapped within its block
• Figure 3: A block graph, its corresponding tree representation and $\mathcal{T}(G)$.
• Figure 4: A block graph $G$ and its relevant subgraphs
• Figure 5: Effect of the parameters $cap(C[p])$ and $ua_G(p)$ on reachability

Theorems & Definitions (62)

• Definition 1
• Definition 2
• Definition 3
• proof
• Definition 4
• Definition 5
• Lemma 1
• proof
• Lemma 2
• proof