12 research outputs found
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension
is a complete isomorphism test for the class of all graphs of rank
width at most . Rank width is a graph invariant that, similarly to tree
width, measures the width of a certain style of hierarchical decomposition of
graphs; it is equivalent to clique width. It was known that isomorphism of
graphs of rank width is decidable in polynomial time (Grohe and Schweitzer,
FOCS 2015), but the best previously known algorithm has a running time
for a non-elementary function . Our result yields an isomorphism
test for graphs of rank width running in time . Another
consequence of our result is the first polynomial time canonisation algorithm
for graphs of bounded rank width. Our second main result is that fixed-point
logic with counting captures polynomial time on all graph classes of bounded
rank width.Comment: 32 page
Computing with Tangles
Tangles of graphs have been introduced by Robertson and Seymour in the
context of their graph minor theory. Tangles may be viewed as describing
"k-connected components" of a graph (though in a twisted way). They play an
important role in graph minor theory. An interesting aspect of tangles is that
they cannot only be defined for graphs, but more generally for arbitrary
connectivity functions (that is, integer-valued submodular and symmetric set
functions).
However, tangles are difficult to deal with algorithmically. To start with,
it is unclear how to represent them, because they are families of separations
and as such may be exponentially large. Our first contribution is a data
structure for representing and accessing all tangles of a graph up to some
fixed order.
Using this data structure, we can prove an algorithmic version of a very
general structure theorem due to Carmesin, Diestel, Harman and Hundertmark (for
graphs) and Hundertmark (for arbitrary connectivity functions) that yields a
canonical tree decomposition whose parts correspond to the maximal tangles.
(This may be viewed as a generalisation of the decomposition of a graph into
its 3-connected components.
Recommended from our members
Graph Theory
This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results