Recognizing Label-Connected Graphs

In classical gossiping, each participant initially knows a distinct piece of information. Whenever two participants call each other, they exchange everything they know. We study Gossiping: given an undirected multigraph whose vertices represent participants and whose edges represent available calls, decide whether the calls can be ordered so that every participant learns all information, using each edge at most once. Equivalently, can one time label per edge make the graph temporally connected by strict temporal paths? This recognition problem is NP-complete; we study its parameterized complexity. After linear-time preprocessing, we give an exact $O(3^{n/3}n^3)$-time algorithm and a deterministic $2^{O(tw)}n$-time algorithm, where $n$ is the number of vertices and $tw$ is the treewidth of the underlying simple graph. We introduce a Tree Contraction Rule that extends the treewidth algorithm to single-exponential algorithms for parameters that may remain small on dense graphs, such as tree-independence number and clique-width, without requiring a decomposition as input. For the excess $δ=m-(2n-4)$ over the minimum number of calls, where $m$ is the number of edges, we give a randomized $2^{O(δ)}n^3$-time algorithm with one-sided error. Its main subroutine solves Rooted Independent Cycle: finding a cycle through four prescribed vertices whose edges are independent in a represented linear matroid. We solve this problem deterministically in $2^{O(r)}m$ arithmetic operations over the finite field representing the matroid, where $r$ is its rank. Gossiping is polynomial-time solvable at maximum degree four, counting parallel edges, but NP-complete on planar simple graphs of maximum degree five. The treewidth algorithm yields a $2^{O(\sqrt n)}$-time algorithm on planar graphs.