35 research outputs found
Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration
We continue research into a well-studied family of problems that ask whether
the vertices of a graph can be partitioned into sets and~, where is
an independent set and induces a graph from some specified graph class
. We let be the class of -degenerate graphs. This
problem is known to be polynomial-time solvable if (bipartite graphs) and
NP-complete if (near-bipartite graphs) even for graphs of maximum degree
. Yang and Yuan [DM, 2006] showed that the case is polynomial-time
solvable for graphs of maximum degree . This also follows from a result of
Catlin and Lai [DM, 1995]. We consider graphs of maximum degree on
vertices. We show how to find and in time for , and in
time for . Together, these results provide an algorithmic
version of a result of Catlin [JCTB, 1979] and also provide an algorithmic
version of a generalization of Brook's Theorem, which was proven in a more
general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007].
Moreover, the two results enable us to complete the complexity classification
of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex
colouring reconfiguration graph between two given -colourings of a graph
of maximum degree
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric
Computing Graph Roots Without Short Cycles
Graph G is the square of graph H if two vertices x, y have an edge in G if
and only if x, y are of distance at most two in H. Given H it is easy to
compute its square H2, however Motwani and Sudan proved that it is NP-complete
to determine if a given graph G is the square of some graph H (of girth 3). In
this paper we consider the characterization and recognition problems of graphs
that are squares of graphs of small girth, i.e. to determine if G = H2 for some
graph H of small girth. The main results are the following. - There is a graph
theoretical characterization for graphs that are squares of some graph of girth
at least 7. A corollary is that if a graph G has a square root H of girth at
least 7 then H is unique up to isomorphism. - There is a polynomial time
algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is
NP-complete to recognize if G = H2 for some graph H of girth 4. These results
almost provide a dichotomy theorem for the complexity of the recognition
problem in terms of girth of the square roots. The algorithmic and graph
theoretical results generalize previous results on tree square roots, and
provide polynomial time algorithms to compute a graph square root of small
girth if it exists. Some open questions and conjectures will also be discussed
Boundary classes for graph problems involving non-local properties
We continue the study of boundary classes for NP-hard problems and focus on seven NP-hard graph problems involving non-local properties: HAMILTONIAN CYCLE, HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE, HAMILTONIAN PATH, FEEDBACK VERTEX SET, CONNECTED VERTEX COVER, CONNECTED DOMINATING SET and GRAPH VCCON DIMENSION. Our main result is the determination of the first boundary class for FEEDBACK VERTEX SET. We also determine boundary classes for HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE and HAMILTONIAN PATH and give some insights on the structure of some boundary classes for the remaining problems
Complexity Framework for Forbidden Subgraphs
For any finite set H={H1,…,Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,…,Hp as a subgraph. Similar to known meta-classifications for the minor and topological minor relations, we give a meta-classification for the subgraph relation. Our framework classifies if problems are "efficiently solvable" or "computationally hard" for H-subgraph-free graphs. The conditions are that the problem should be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness is preserved under edge subdivision. We show that all problems satisfying these conditions are efficiently solvable if H contains a disjoint union of one or more paths and subdivided claws, and are computationally hard otherwise. To illustrate the broad applicability of our framework, we study partitioning, covering and packing problems, network design problems and width parameter problems. We apply the framework to obtain a dichotomy between polynomial-time solvability and NP-completeness. For other problems we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). Along the way we unify and strengthen known results from the literature
Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration
We continue research into a well-studied family of problems that ask whether the vertices of a given graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We consider the case where G is the class of k-degenerate graphs. This problem is known to be polynomial-time solvable if k = 0 (recognition of bipartite graphs), but NP-complete if k = 1 (near-bipartite graphs) even for graphs of maximum degree 4. Yang and Yuan [DM, 2006] showed that the k = 1 case is polynomial-time solvable for graphs of maximum degree 3. This also follows from a result of Catlin and Lai [DM, 1995]. We study the general k ≥ 1 case for n-vertex graphs of maximum degree k + 2 We show how to find A and B in O(n) time for k = 1, and in O(n 2 ) time for k ≥ 2. Together, these results provide an algorithmic version of a result of Catlin [JCTB, 1979] and also provide an algorithmic version of a generalization of Brook’s Theorem, proved by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007]. The results also enable us to solve an open problem of Feghali et al. [JGT, 2016]. For a given graph G and positive integer `, the vertex colouring reconfiguration graph of G has as its vertex set the set of `-colourings of G and contains an edge between each pair of colourings that differ on exactly on vertex. We complete the complexity classification of the problem of finding a path in the reconfiguration graph between two given `-colourings of a given graph of maximum degree k
Star Colouring of Bounded Degree Graphs and Regular Graphs
A -star colouring of a graph is a function
such that for every edge of
, and every bicoloured connected subgraph of is a star. The star
chromatic number of , , is the least integer such that is
-star colourable. We prove that for
every -regular graph with . We reveal the structure and
properties of even-degree regular graphs that attain this lower bound. The
structure of such graphs is linked with a certain type of Eulerian
orientations of . Moreover, this structure can be expressed in the LC-VSP
framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by
an FPT algorithm with the parameter either treewidth, cliquewidth, or
rankwidth. We prove that for , a -regular graph is
-star colourable only if is divisible by . For
each and divisible by , we construct a -regular
Hamiltonian graph on vertices which is -star colourable.
The problem -STAR COLOURABILITY takes a graph as input and asks
whether is -star colourable. We prove that 3-STAR COLOURABILITY is
NP-complete for planar bipartite graphs of maximum degree three and arbitrarily
large girth. Besides, it is coNP-hard to test whether a bipartite graph of
maximum degree eight has a unique 3-star colouring up to colour swaps. For
, -STAR COLOURABILITY of bipartite graphs of maximum degree is
NP-complete, and does not even admit a -time algorithm unless ETH
fails