Jisu Jeong

(1, k)-Coloring of Graphs with Girth at Least Five on a Surface

A graph is (d1,...,dr)-colorable if its vertex set can be partitioned into r sets V1,...,Vr so that the maximum degree of the graph induced by Vi is at most di for each i∈{1,...,r}. For a given pair (g,d1), the question of determining the minimum d2=d2(g,d1) such that planar graphs with girth at least g are (d1,d2)-colorable has attracted much interest. The finiteness of d2(g,d1) was known for all cases except when (g,d1)=(5,1). Montassier and Ochem explicitly asked if d2(5, 1) is finite. We answer this question in the affirmative with d2(5,1)≤10; namely, we prove that all planar graphs with girth at least five are (1, 10)-colorable. Moreover, our proof extends to the statement that for any surface S of Euler genus γ, there exists a K=K(γ) where graphs with girth at least five that are embeddable on S are (1, K)-colorable. On the other hand, there is no finite k where planar graphs (and thus embeddable on any surface) with girth at least five are (0, k)-colorable.

Maximum matching width: New characterizations and a fast algorithm for dominating set

Abstract A graph of treewidth k has a representation by subtrees of a ternary tree, with subtrees of adjacent vertices sharing a tree node, and any tree node sharing at most k + 1 subtrees. Likewise for branchwidth, but with a shift to the edges of the tree rather than the nodes. In this paper we show that the mm-width of a graph – maximum matching width – combines aspects of both these representations, targeting tree nodes for adjacency and tree edges for the parameter value. The proof of this new characterization of mm-width is based on a definition of canonical minimum vertex covers of bipartite graphs. We show that these behave in a monotone way along branch decompositions over the vertex set of a graph. We use these representations to compare mm-width with treewidth and branchwidth, and also to give another new characterization of mm-width, by subgraphs of chordal graphs. We prove that given a graph G and a branch decomposition of maximum matching width k we can solve the Minimum Dominating Set Problem in time O ∗ ( 8 k ) , thereby beating O ∗ ( 3 tw ( G ) ) whenever tw ( G ) > log 3 8 × k ≈ 1 . 893 k . Note that mmw ( G ) ≤ tw ( G ) + 1 ≤ 3 mmw ( G ) and these inequalities are tight. Given only the graph G and using the best known algorithms to find decompositions, maximum matching width will be better for Minimum Dominating Set whenever tw ( G ) > 1 . 549 × mmw ( G ) .

Excluded vertex-minors for graphs of linear rank-width at most k.

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite set \mathcal{O}_k of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in \mathcal{O}_k. However, no attempts have been made to bound the number of graphs in \mathcal{O}_k for k >= 2. We construct, for each k, 2^{\Omega(3^k)} pairwise locally non-equivalent graphs that are excluded vertex-minors for graphs of linear rank-width at most k. Therefore the number of graphs in \mathcal{O}_k is at least double exponential.

Characterizing graphs of maximum matching width at most 2

The maximum matching width is a width-parameter that is defined on a branch-decomposition over the vertex set of a graph. The size of a maximum matching in the bipartite graph is used as a cut-function. In this paper, we characterize the graphs of maximum matching width at most 2 using the minor obstruction set. Also, we compute the exact value of the maximum matching width of a grid.

Maximum Matching Width: New Characterizations and a Fast Algorithm for Dominating Set.

We give alternative definitions for maximum matching width, e.g. a graph

Excluded vertex-minors for graphs of linear rank-width at most k

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Constructive algorithm for path-width of matroids

has

The “Art of Trellis Decoding” Is Fixed-Parameter Tractable

\operatorname{mmw}(G) \leq k

Computing Small Pivot-Minors.

if and only if it is a subgraph of a chordal graph

Finding Branch-Decompositions of Matroids, Hypergraphs, and More.

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