In graph theory, the tree-depth of a connected undirected graph is a numerical invariant of , the minimum height of a Trémaux tree for a supergraph of . This invariant and its close relatives have gone under many different names in the literature, including vertex ranking number, ordered chromatic number, and minimum elimination tree height; it is also closely related to the cycle rank of directed graphs and the star height of regular languages. Intuitively, where the treewidth of a graph measures how far it is from being a tree, this parameter measures how far a graph is from being a star.
The tree-depth of a graph may be defined as the minimum height of a forest with the property that every edge of connects a pair of nodes that have an ancestor-descendant relationship to each other in . If is connected, this forest must be a single tree; it need not be a subgraph of , but if it is, it is a Trémaux tree for .
The set of ancestor-descendant pairs in forms a trivially perfect graph, and the height of is the size of the largest clique in this graph. Thus, the tree-depth may alternatively be defined as the size of the largest clique in a trivially perfect supergraph of , mirroring the definition of treewidth as one less than the size of the largest clique in a chordal supergraph of .
Another definition is the following:
where is the set of vertices of and the are the connected components of . This definition mirrors the definition of cycle rank of directed graphs, which uses strong connectivity and strongly connected components in place of undirected connectivity and connected components.
Tree-depth may also be defined using a form of graph coloring. A centered coloring of a graph is a coloring of its vertices with the property that every connected induced subgraph has a color that appears exactly once. Then, the tree-depth is the minimum number of colors in a centered coloring of the given graph. If is a forest of height with the property that every edge of connects an ancestor and a descendant in , then a centered coloring of using colors may be obtained by coloring each vertex by its distance from the root of its tree in .
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The course aims to introduce the basic concepts and results of modern Graph Theory with special emphasis on those topics and techniques that have proved to be applicable in theoretical computer scienc
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number that measures how much the path was thickened to form G. More formally, a path-decomposition is a sequence of subsets of vertices of G such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition.
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph. While graph drawing and graph representation are valid topics in graph theory, in order to focus only on the abstract structure of graphs, a graph property is defined to be a property preserved under all possible isomorphisms of a graph. In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph.
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