6 research outputs found
Computing graph gonality is hard
There are several notions of gonality for graphs. The divisorial gonality
dgon(G) of a graph G is the smallest degree of a divisor of positive rank in
the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the
minimum degree of a finite harmonic morphism from a refinement of G to a tree,
as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and
sgon(G) are NP-hard by a reduction from the maximum independent set problem and
the vertex cover problem, respectively. Both constructions show that computing
gonality is moreover APX-hard.Comment: The previous version only dealt with hardness of the divisorial
gonality. The current version also shows hardness of stable gonality and
discusses the relation between the two graph parameter
Constructing tree decompositions of graphs with bounded gonality
In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality
Recognizing hyperelliptic graphs in polynomial time
Recently, a new set of multigraph parameters was defined, called
"gonalities". Gonality bears some similarity to treewidth, and is a relevant
graph parameter for problems in number theory and multigraph algorithms.
Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic
graphs" (multigraphs of gonality 2) and provide a safe and complete sets of
reduction rules for such multigraphs, showing that for three of the flavors of
gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n
is the number of vertices and m the number of edges of the multigraph.Comment: 33 pages, 8 figure
Constructing tree decompositions of graphs with bounded gonality
In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality