Edge degeneracy: Algorithmic and structural results

Stratis Limnios, Christophe Paul, Joanny Perret, Dimitrios M. Thilikos

Research output: Contribution to journalArticlepeer-review


We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges (s can be seen as the speed of the robber). Both parts have complete knowledge of the opponent's moves and the cops win when they occupy all edges incident to the robbers position. We introduce the capture cost on G against a robber of speed s. This defines a hierarchy of invariants, namely δe1e2,…,δe, where δe is an edge-analogue of the admissibility graph invariant, namely the edge-admissibility of a graph. We prove that the problem asking whether δes(G)≤k, is polynomially solvable when s∈{1,2,3,∞} while, otherwise, it is NP-complete. Our main result is a structural theorem for graphs of bounded edge-admissibility. We prove that every graph of edge-admissibility at most k can be constructed using (≤k)-edge-sums, starting from graphs whose all vertices, except possibly from one, have degree at most k. Our structural result is approximately tight in the sense that graphs generated by this construction always have edge-admissibility at most 2k−1. Our proofs are based on a precise structural characterization of the graphs that do not contain θr as an immersion, where θr is the graph on two vertices and r parallel edges.

Original languageEnglish (US)
Pages (from-to)164-175
Number of pages12
JournalTheoretical Computer Science
StatePublished - Nov 2 2020


  • Cops and robber games
  • Graph admissibility
  • Graph decomposition theorems
  • Graph degeneracy
  • Graph searching

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


Dive into the research topics of 'Edge degeneracy: Algorithmic and structural results'. Together they form a unique fingerprint.

Cite this