Vertex Separators and low tree-width k-coloring
ثبت نشده
چکیده
Given a graph G(V,E) and a set of vertices S ⊂ V , an S-flap is the set of vertices in a connected component of the graph induced on V \ S. A set S is a vertex separator if no S-flap has more than n/2 vertices. Lipton and Tarjan showed that every planar graph has a separator of size O( √ n). This was later generalized by Gilbert, Hutchinson and Tarjan to any graph embeddable on a surface of bounded genus. This was further generalized by Alon, Seymour and Thomas to any family of graphs that excludes some fixed (arbitrary) subgraph H as a minor. Their proof (like all previous proofs) is constructive – it provides a polynomial time algorithm that finds the desired separator.
منابع مشابه
Testing Superperfection of k-Trees
An interval coloring of a weighted graph with non-negative weights, maps each vertex onto an open interval on the real line with width equal to the weight of the vertex, such that adjacent vertices are mapped to disjoint intervals. The total width of an interval coloring is defined as the width of the union of the intervals. The interval chromatic number of a weighted graph is the least total w...
متن کاملOn the Tractability of (k, i)-Coloring
In an undirected graph, a proper (k, i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k, i)-coloring problem is to compute the minimum number of colors required for a proper (k, i)coloring. This is a generalization of the classic graph coloring problem. Majumdar et. al. [CALDAM 2017] studied this problem and ...
متن کاملEdge-coloring Vertex-weightings of Graphs
Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...
متن کاملGraphs with complete minimal k-vertex separators
G. A. Dirac characterized chordal graphs as those in which minimal vertex separators always induce complete subgraphs. I generalize a traditional (2-)vertex separator to a k-vertex separator — meaning a set S of vertices whose removal puts k independent vertices into k separate components. Generalizing Dirac’s theorem, the {P5, 2P3}-free chordal graphs are the graphs in which minimal k-separato...
متن کاملOn tree width, bramble size, and expansion
A bramble in a graph G is a family of connected subgraphs of G such that any two of these subgraphs have a nonempty intersection or are joined by an edge. The order of a bramble is the least number of vertices required to cover every subgraph in the bramble. Seymour and Thomas [8] proved that the maximum order of a bramble in a graph is precisely the tree width of the graph plus one. We prove t...
متن کامل