We present a simple linear-time algorithm that outputs an Eulerian tour of an undirected multigraph with $n$ vertices and $m$ edges, if one exists, in $O(m)$ time and using $O(n)$ words of working memory. The input is given as read-only adjacency lists, and the output is written to an append-only stream in traversal order. Our algorithm first finds a sparse spanning circuit (a skeleton), then traverses the circuit step-by-step, repeatedly outputting further circuits rooted at the current vertex. This solves a problem left open by Ismaili Alaoui, Plump, and Wild (SOSA 2026): their space-efficient variant of Hierholzer's algorithm handles general directed multigraphs, but it is unclear how to generalize it to general undirected multigraphs. Our result completes the picture in the read-only model for space-efficient output of Eulerian tours.
It is shown that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order, and under the order-dependent model, where bidirectional Dijkstra is not instance-optimal.
Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup et al.· 1 citation
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completi...
It is proved that Minimal-to-Maximal Conversion Search is in fact not output-polynomial and the lower bound construction motivates a more detailed analysis of how certain heuristic choices in the algorithm design affect the running time.
This paper provides the first truly linear-time approximation scheme for the Densest Subgraph Problem, and uses assignments arising from a flow-based formulation together with a structural carving lemma to progressively carve "sparse" parts of the graph while nearly preserving the densest subgraph.
It is proved that the classical cut property always produces a minimum spanning tree of a connected graph, and may be useful for large weighted networks such as communication networks, wiring connections, and transportation networks.
H. Bhapkar, Rezwan Ul Shaban, S. Mir et al.· Journal of the Nigerian Soci...· 0 citations
A randomized algorithm is designed that runs in $\tilde{O}(n^2)$ time and, with high probability, guarantees a 2-approximation for all pairs at distance at least $c$, where $c \ge 0$ is a constant.
