biology

Rooted graph

Rooted graph is a biology topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Rooted graph rather than just read about it. In short: In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots.

Key takeaways

Rooted graph belongs to biology; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rooted graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rooted graph from memory before moving on to harder problems.

Reference excerpt

In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots.

Rooted graphs may also be known (depending on their application) as pointed graphs or flow graphs. In some of the applications of these graphs, there is an additional requirement that the whole graph be reachable from the root vertex.

Variations In topological graph theory, the notion of a rooted graph may be extended to consider multiple vertices or multiple edges as roots. The former are sometimes called vertex-rooted graphs in order to distinguish them from edge-rooted graphs in this context. Graphs with multiple nodes designated as roots are also of some interest in combinatorics, in the area of random graphs. These graphs are also called multiply rooted graphs. The terms rooted directed graph or rooted digraph also see variation in definitions. The obvious transplant is to consider a digraph rooted by identifying a particular node as root. However, in computer science, these terms commonly refer to a narrower notion; namely, a rooted directed graph is a digraph with a distinguished node r, such that there is a directed path from r to any node other than r. Authors who give the more general definition may refer to graphs meeting the narrower definition as connected rooted digraphs or accessible rooted graphs (see § Set theory). The Art of Computer Programming defines rooted digraphs slightly more broadly, namely, a directed graph is called rooted if it has at least one node that can reach all the other nodes. Knuth notes that the notion thus defined is a sort of intermediate between the notions of strongly connected and connected digraph.

Applications

Flow graphs In computer science, rooted graphs in which the root vertex can reach all other vertices are called flow graphs or flowgraphs. Sometimes an additional restriction is added specifying that a flow graph must have a single exit (sink) vertex. Flow graphs may be viewed as abstractions of flow charts, with the non-structural elements (node contents and types) removed. Perhaps the best known sub-class of flow graphs are control-flow graphs, used in compilers and program analysis. An arbitrary flow graph may be converted to a control-flow graph by performing an edge contraction on every edge that is the only outgoing edge from its source and the only incoming edge into its target. Another type of flow graph commonly used is the call graph, in which nodes correspond to entire subroutines. The general notion of flow graph has been called program graph, but the same term has also been used to denote only control-flow graphs. Flow graphs have also been called unlabeled flowgraphs and proper flowgraphs. These graphs are sometimes used in software testing. When required to have a single exit, flow graphs have two properties not shared with directed graphs in general: flow graphs can be nested, which is the equivalent of a subroutine call (although there is no notion of passing parameters), and flow graphs can also be sequenced, which is the equivalent of sequential execution of two pieces of code. Prime flow graphs are defined as flow graphs that cannot be decomposed via nesting or sequencing using a chosen pattern of subgraphs, for example the primitives of structured programming. Theoretical research has been done on determining, for example, the proportion of prime flow graphs given a chosen set of graphs.

Set theory Peter Aczel has used rooted directed graphs such that every node is reachable from the root (which he calls accessible pointed graphs) to formulate Aczel's anti-foundation axiom in non-well-founded set theory. In this context, each vertex of an accessible pointed graph models a (non-well-founded) set within Aczel's (non-well-founded) set theory, and an arc from a vertex v to a vertex w models that v is an element of w. Aczel's anti-foundation axiom states that every accessible pointed graph models a family of (non-well-founded) sets in this way.

Combinatorial game theory Any combinatorial game, can be associated with a rooted directed graph whose vertices are game positions, whose edges are moves, and whose root is the starting position of the game. This graph is important in the study of game complexity, where the state-space complexity is the number of vertices in the graph.

Combinatorial enumeration The number of rooted undirected graphs for 1, 2, ... nodes is 1, 2, 6, 20, 90, 544, ... (sequence A000666 in the OEIS).

Related concepts A special case of interest are rooted trees, the trees with a distinguished root vertex. If the directed paths from the root in the rooted digraph are additionally restricted to be unique, then the notion obtained is that of (rooted) arborescence—the directed-graph equivalent of a rooted tree. A rooted graph contains an arborescence with the same root if and only if the whole graph can be reached from the root, and computer scientists have studied algorithmic problems of finding optimal arborescences. Rooted graphs may be combined using the rooted product of graphs.

See also k-vertex-connected graph pointed set

References

Further reading McMahon, Elizabeth W. (1993), "On the greedoid polynomial for rooted graphs and rooted digraphs", Journal of Graph Theory, 17 (3): 433–442, doi:10.1002/jgt.3190170316 Gordon, Gary (2001), "A characteristic polynomial for rooted graphs and rooted digraphs", Discrete Mathematics, 232 (1–3): 19–33, doi:10.1016/S0012-365X(00)00186-2

External links Weisstein, Eric W., "Rooted Graph", MathWorld

Illustrations

Worked examples

Example 1 — a first encounter with Rooted graph

Start with the simplest possible case. Write down what Rooted graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In biology, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rooted graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rooted graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rooted graph

In research: Rooted graph appears in biology research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rooted graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rooted graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Extensions and generalizations of graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Rooted graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Rooted graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rooted graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rooted graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rooted graph in simple terms?

Why does Rooted graph matter?

Because it connects several biology ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rooted graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rooted graph.