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| In [[graph theory]], the '''tree-depth''' of a [[connected graph|connected]] [[undirected graph]] ''G'' is a numerical invariant of ''G'', the minimum height of a [[Trémaux tree]] for a [[Glossary of graph theory#Subgraphs|supergraph]] of ''G''. 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 graph]]s and the [[star height]] of [[regular language]]s.<ref>{{harvtxt|Bodlaender|Deogun|Jansen|Kloks|1998}}; {{harvtxt|Rossman|2008}}; {{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 116.</ref> Intuitively, where the [[treewidth]] graph width parameter measures how far a graph is from being a tree, this parameter measures how far a graph is from being a star.
| | <br><br>A terrific knife - or a whole set of great knives - can make your kitchen prep a lot less complicated, and may possibly even make cooking exciting once more. You fully grasp the worth of an award-winning illustrator when you see Knife Abilities Illustrated. Are you a cook? There are times when you are tempted to acquire a new set of knives when the issue is just the sharpness of the cutlery. To lessen your costs for cutlery, it would be vital to check on your set and sharpened them after in a when. It may well be much less expensive to invest in a set than individually. If you do substantially cooking at all, you are going to use not one, but various sorts of knives. Each and every of these kitchen utensils has a precise function, like chopping, slicing, or filleting meals. I am created of plastic and metal ones.<br><br>Serving Any Cheese: Boska Havana Cheese Set Utilized by: Charlotte Kamin, owner-monger, Bedford Cheese Shop Buy at: Boska, $15 Kamin, as well, recommends most any knife from Boska. This brightly colored, dishwasher-friendly stainless-steel set will suit casual cheese lovers it comes with a versatile slicer a spade knife, for tough cheeses and a forkedfor holding, serving, and crumbling cheeses plus, the set consists of a serving board. A good set of cookware and bake ware.<br><br>I really feel these Wusthof knives are the most productive developed of the five lines I have examined. They've joined with each other the most helpful functions of Japanese kitchen knives with each other with the bolsterless design and style and style and intensely steep angle about the edge. These chef knives are subsequently nicely balanced, lightweight and have absolutely exceptional sharpness. Use an older pitcher that is bottom-heavy and tapered toward the top rated.<br><br>A further colourful set to inject that vibrancy into your kitchen, this knife block set contains the 5 knives you are going to locate your self needing the most. This set is perfect for the day-to-day home cook who does not want to fork out for specialist excellent but likes a knife to do its job quite effectively. The Twin Pro S series knives are forged, on the other hand, their handles are welded and not forged.<br><br>The utility knife is advertised for working with on soft roasts, sandwiches and so on. I tend to use it a fair bit but then I rather like the smaller size but a very good bread knife could be a better compromise in a set like this. The set could be improved with the addition of a bread knife and a sharpening tool - if those have been included I feel this would make an superb set for any kitchen. If this you then stamped knives would be finest.<br><br>According to Customer Reports, "Although the top rated-rated knives are forged, stamped knives are capable of really fantastic performance." The non-profit consumer education organization points out that stamped knives do not have a bolster and a heel, which do add worth to the user encounter but those attributes most likely would not be missed by most people today. Take a look at their internet web-site for a very good "crash course" in understanding knife terminology and ideas on effective use and care of kitchen knives. Use a knife block.<br><br>As for slicing cooked meat, there is a slicing knife for it. A slicing knife has a extended and narrow blade in contrast to those knives with a scalloped edge which are excellent for [http://thesaurus.com/browse/slicing+soft slicing soft] food like tomatoes, bread and cakes. The paring knife is a smaller knife with a quick pointed blade that makes it uncomplicated to manage and use. Effectively the uses of knives are not only restricted to the kitchen as they are made use of elsewhere in the home, mostly on the dinner table. They are a ought to have in my kitchen.<br><br>It can be secured to the underside of an upper cupboard, thus producing knives speedily accessible without the need of taking up any counter space. Mounted blocks are generally smaller sized than counter-prime blocks, and hold only your most often applied knives. If your meal is ordinarily light, like extra on vegetables and fruits, then you should decide on lighter kitchen knives. If you are you looking for more in regards to [http://www.thebestkitchenknivesreviews.com/best-japanese-knives-chef-models-review/ Greatest Rated Bread Knife] stop by our web-page. A high-quality set of knives is a terrific productivity booster.<br><br>Envy Spiral Slicer is an item that comes with really sharp and challenging Japanese blades that have been specifically made in order to stop corrosion. IPerfect Kitchen tends to make confident all prospects of a stainless steel spiral slicer that could be conveniently washed, either by hand or utilizing the dishwashing machine, with no going via long refines that demand sophisticated meal soap. Place a damp cloth on the kitchen bench. |
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| ==Definitions==
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| The tree-depth of a graph ''G'' may be defined as the minimum height of a [[forest (graph theory)|forest]] ''F'' with the property that every edge of ''G'' connects a pair of nodes that have an ancestor-descendant relationship to each other in ''F''.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Definition 6.1, p. 115.</ref> If ''G'' is connected, this forest must be a single tree; it need not be a subgraph of ''G'', but if it is, it is a [[Trémaux tree]] for ''G''.
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| The set of ancestor-descendant pairs in ''F'' forms a [[trivially perfect graph]], and the height of ''F'' is the size of the largest [[clique (graph theory)|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 ''G'', mirroring the definition of [[treewidth]] as one less than the size of the largest clique in a [[chordal graph|chordal]] supergraph of ''G''.
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| Another definition is the following:
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| <math>td(G)=\begin{cases}1, & \text{if }|G|=1;\\
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| 1+\min_{v\in V} td(G-v), & \text{if }G\text{ is connected and }|G|>1;\\
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| \max_{i} td(G_i), &\text{otherwise};
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| \end{cases}</math>
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| where the <math>G_i</math> are the connected components of ''G''.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Lemma 6.1, p. 117.</ref>
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| 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 ''F'' is a forest of height ''d'' with the property that every edge of ''G'' connects an ancestor and a descendant in the tree, then a centered coloring of ''G'' using ''d'' colors may be obtained by coloring each vertex by its distance from the root of its tree in ''F''.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Section 6.5, "Centered Colorings", pp. 125–128.</ref>
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| Finally, one can define this in terms of a [[pebble game]], or more precisely as a [[Pursuit-evasion|cops and robber]] game. Consider the following game, played on an undirected graph. There are two players, a robber and a cop. The robber has one pebble he can move along the edges of the given graph. The cop has an unlimited number of pebbles, but she wants to minimize the amount of pebbles she uses. The cop cannot move a pebble after it's been placed on the graph. The game proceeds as follows. The robber places his pebble. The cop then announces where she wants to place a new cop pebble. The robber can then move his pebble along edges, but not through occupied vertices. The game is over when the cop player places a pebble on top of the robber pebble. The tree-depth of the given graph is the minimum number of pebbles needed by the cop to guarantee a win.<ref>{{harvtxt|Gruber|Holzer|2008}}, Theorem 5, {{harvtxt|Hunter|2011}}, Main Theorem.</ref> For a star graph, this is 2 (the strategy is to place at the center vertex, forcing the robber to one arm, and then to place the remaining pebble on the robber). For a path, the robber uses a [[binary search]] strategy, which guarantees a ''log n'' number of pebbles needed.
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| ==Examples==
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| [[File:Tree-depth.svg|thumb|360px|The tree-depths of the [[complete graph]] ''K''<sub>4</sub> and the [[complete bipartite graph]] ''K''<sub>3,3</sub> are both four, while the tree-depth of the [[path graph]] ''P''<sub>7</sub> is three.]]
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| The tree-depth of a [[complete graph]] equals its number of vertices, for in this case the only possible forest ''F'' for which every pair of vertices are in an ancestor-descendant relationship is a single path. Similarly, the tree-depth of a [[complete bipartite graph]] ''K''<sub>''x'',''y''</sub> is min(''x'',''y'') + 1, for whatever nodes are placed at the leaves of the forest ''F'' must have at least min(''x'',''y'') ancestors in ''F''. A forest achieving this min(''x'',''y'') + 1 bound may be constructed by forming a path for the smaller side of the bipartition, with each vertex on the larger side of the bipartition forming a leaf in ''F'' connected to the bottom vertex of this path.
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| The tree-depth of a path with ''n'' vertices is exactly <math>\lceil\log_2(n+1)\rceil</math>. A forest ''F'' representing this path with this depth may be formed by placing the midpoint of the path as the root of ''F'' and recursing within the two smaller paths on either side of it.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Formula 6.2, p. 117.</ref>
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| ==Depth of trees and relation to treewidth==
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| Any ''n''-vertex [[tree (graph theory)|forest]] has tree-depth O(log ''n''). For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2''n''/3 vertices each. By recursively partitioning each of these two subforests, we can easily derive a logarithmic upper bound on the tree-depth. The same technique, applied to a [[tree decomposition]] of a graph, shows that, if the [[treewidth]] of an ''n''-vertex graph ''G'' is ''t'', then the tree-depth of ''G'' is O(''t'' log ''n'').<ref>{{harvtxt|Bodlaender|Gilbert|Hafsteinsson|Kloks|1995}}; {{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Corollary 6.1, p. 124.</ref> Since [[outerplanar graph]]s, [[series-parallel graph]]s, and [[Halin graph]]s all have bounded treewidth, they all also have at most logarithmic tree-depth.
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| In the other direction, the treewidth of a graph is at most equal to its tree-depth. More precisely, the treewidth is at most the [[pathwidth]], which is at most one less than the tree-depth.<ref>{{harvtxt|Bodlaender|Gilbert|Hafsteinsson|Kloks|1995}}; {{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 123.</ref>
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| ==Graph minors==
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| A [[minor (graph theory)|minor]] of a graph ''G'' is another graph formed from a subgraph of ''G'' by contracting some of its edges. Tree-depth is monotonic under minors: every minor of a graph ''G'' has tree-depth at most equal to the tree-depth of ''G'' itself.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Lemma 6.2, p. 117.</ref> Thus, by the [[Robertson–Seymour theorem]], for every fixed ''d'' the set of graphs with tree-depth at most ''d'' has a finite set of [[forbidden graph characterization|forbidden minors]].
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| If ''C'' is a class of graphs closed under taking graph minors, then the graphs in ''C'' have tree-depth <math>O(1)</math> if and only if ''C'' does not include all the [[path graph]]s.<ref name="no12-prop64">{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Proposition 6.4, p. 122.</ref>
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| ==Induced subgraphs==
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| As well as behaving well under graph minors, tree-depth has close connections to the theory of [[induced subgraph]]s of a graph. Within the class of graphs that have tree-depth at most ''d'' (for any fixed integer ''d''), the relation of being an induced subgraph forms a [[well-quasi-ordering]].<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, Lemma 6.13, p. 137.</ref> The basic idea of the proof that this relation is a well-quasi-ordering is to use induction on ''d''; the forests of height ''d'' may be interpreted as sequences of forests of height ''d'' − 1 (formed by deleting the roots of the trees in the height-''d'' forest) and [[Higman's lemma]] can be used together with the induction hypothesis to show that these sequences are well-quasi-ordered.
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| Well-quasi-ordering implies that any property of graphs that is monotonic with respect to induced subgraphs has finitely many forbidden induced subgraphs, and therefore may be tested in polynomial time on graphs of bounded tree-depth. The graphs with tree-depth at most ''d'' themselves also have a finite set of forbidden induced subgraphs.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 138. Figure 6.6 on p. 139 shows the 14 forbidden subgraphs for graphs of tree-depth at most three, credited to the 2007 Ph.D. thesis of Z. Dvořák.</ref>
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| If ''C'' is a class of graphs with bounded [[degeneracy (graph theory)|degeneracy]], the graphs in ''C'' have bounded tree-depth if and only if there is a path graph that cannot occur as an induced subgraph of a graph in ''C''.<ref name="no12-prop64"/>
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| ==Complexity==
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| Computing the tree-depth is computationally hard: the corresponding decision problem is [[NP-complete]].<ref name="p88">{{harvtxt|Pothen|1988}}.</ref> The problem remains NP-complete for [[complement (graph theory)|complement]]s of [[bipartite graph]]s,<ref name="p88" for [[bipartite graph]]s {{harv|Bodlaender|Deogun|Jansen|Kloks|1998}}, as well as for [[chordal graph]]s.<ref>{{harvtxt|Dereniowski|Nadolski|2006}}.</ref>
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| On the positive side, the problem is solvable in [[polynomial time]] on interval graphs,<ref>{{harvtxt|Aspvall|Heggernes|1994}}.</ref> as well as on permutation, trapezoid, circular-arc, circular permutation graphs, and cocomparability graphs of bounded dimension.<ref>{{harvtxt|Deogun|Kloks|Kratsch|Müller|1999}}.</ref> For undirected trees, the problem is solvable in linear time.<ref>{{harvtxt|Iyer|Ratliff|Vijayan|1988}}; {{harvtxt|Schäffer|1989}}.</ref>
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| {{harvtxt|Bodlaender|Gilbert|Hafsteinsson|Kloks|1995}} give an [[approximation algorithm]] with approximation ratio <math>O((\log n)^2)</math>.
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| Because tree-depth is monotonic under graph minors, it is [[Parameterized complexity|fixed-parameter tractable]]: there is an algorithm for computing tree-depth whose time is <math>f(d) n^{O(1)}</math>, where ''d'' is the depth of the given graph and ''n'' is its number of vertices. Thus, for every fixed value of ''d'', the problem of testing whether the depth is at most ''d'' can be solved in [[polynomial time]]. More specifically, the dependence on ''n'' in this algorithm can be made linear, by the following method: compute a depth first search tree, and test whether this tree's depth is greater than 2<sup>''d''</sup>. If so, the tree-depth is greater than ''d'' and the problem is solved. If not, the shallow depth first search tree can be used to construct a [[tree decomposition]] with bounded width, and standard [[dynamic programming]] techniques for graphs of bounded treewidth can be used to compute the depth in linear time.<ref>{{harvtxt|Nešetřil|Ossona de Mendez|2012}}, p. 138. A more complicated linear time algorithm based on the planarity of the excluded minors for tree-depth was given earlier by {{harvtxt|Bodlaender|Deogun|Jansen|Kloks|1998}}.</ref>
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| ==Notes==
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| {{reflist|colwidth=30em}}
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| ==References==
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| [[Category:Graph coloring]]
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| [[Category:Graph minor theory]]
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