460 research outputs found
Branch-depth: Generalizing tree-depth of graphs
We present a concept called the branch-depth of a connectivity function, that
generalizes the tree-depth of graphs. Then we prove two theorems showing that
this concept aligns closely with the notions of tree-depth and shrub-depth of
graphs as follows. For a graph and a subset of we let
be the number of vertices incident with an edge in and an
edge in . For a subset of , let be the rank
of the adjacency matrix between and over the binary field.
We prove that a class of graphs has bounded tree-depth if and only if the
corresponding class of functions has bounded branch-depth and
similarly a class of graphs has bounded shrub-depth if and only if the
corresponding class of functions has bounded branch-depth, which we
call the rank-depth of graphs.
Furthermore we investigate various potential generalizations of tree-depth to
matroids and prove that matroids representable over a fixed finite field having
no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure
Branch-depth: Generalizing tree-depth of graphs
We present a concept called the branch-depth of a connectivity function, that
generalizes the tree-depth of graphs. Then we prove two theorems showing that
this concept aligns closely with the notions of tree-depth and shrub-depth of
graphs as follows. For a graph and a subset of we let
be the number of vertices incident with an edge in and an
edge in . For a subset of , let be the rank
of the adjacency matrix between and over the binary field.
We prove that a class of graphs has bounded tree-depth if and only if the
corresponding class of functions has bounded branch-depth and
similarly a class of graphs has bounded shrub-depth if and only if the
corresponding class of functions has bounded branch-depth, which we
call the rank-depth of graphs.
Furthermore we investigate various potential generalizations of tree-depth to
matroids and prove that matroids representable over a fixed finite field having
no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio
Scattered classes of graphs
For a class of graphs equipped with functions defined
on subsets of or , we say that is -scattered with
respect to if there exists a constant such that for every graph
, the domain of can be partitioned into subsets of size
at most so that the union of every collection of the subsets has
value at most . We present structural characterizations of graph classes
that are -scattered with respect to several graph connectivity functions.
In particular, our theorem for cut-rank functions provides a rough structural
characterization of graphs having no vertex-minor, which allows us
to prove that such graphs have bounded linear rank-width.Comment: 42 pages, 5 figures. Adding a new section comparing these concepts
with tree-depth, rank-depth, shrub-depth, modular-width, neighborhood
diversity, et
Graphs of Small Rank-width are Pivot-minors of Graphs of Small Tree-width
We prove that every graph of rank-width is a pivot-minor of a graph of
tree-width at most . We also prove that graphs of rank-width at most 1,
equivalently distance-hereditary graphs, are exactly vertex-minors of trees,
and graphs of linear rank-width at most 1 are precisely vertex-minors of paths.
In addition, we show that bipartite graphs of rank-width at most 1 are exactly
pivot-minors of trees and bipartite graphs of linear rank-width at most 1 are
precisely pivot-minors of paths.Comment: 16 pages, 7 figure
Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k
International audienceEvery minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field , the list contains only finitely many -representable matroids, due to the well-quasi-ordering of -representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these -representable excluded minors in general. We consider the class of matroids of path-width at most for fixed . We prove that for a finite field , every -representable excluded minor for the class of matroids of path-width at most~ has at most elements. We can therefore compute, for any integer and a fixed finite field , the set of -representable excluded minors for the class of matroids of path-width , and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an -represented matroid is at most . We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most has at most vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs
A Polynomial Kernel for 3-Leaf Power Deletion
For a non-negative integer ?, a graph G is an ?-leaf power of a tree T if V(G) is equal to the set of leaves of T, and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most ?. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S ? V(G) of size at most k such that GS is a 3-leaf power of some treeT. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance (G,k) to output an equivalent instance (G\u27,k\u27) such that k\u27? k and G\u27 has at most O(k^14) vertices
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