Maximal independent set¶

In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10620
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Graph Theory → Mathematics

Core Idea¶

Maximal independent set is treated here as the recurring computer science and information systems identity summarized by this source-grounded definition: In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set.

has six different independent sets, shown as the red vertices. In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property.

For example, in the graph , a path with three vertices , , and , and two edges and , the sets and are both maximal independent. The set is independent, but is not maximal independent, because it is a subset of the larger independent set In this same graph, the maximal cliques are the sets and. A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets.

For Maximal independent set, the abstraction is narrower than the article's general subject matter: a positive case must preserve In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in computer science and information systems, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — Any maximal independent set in this graph is formed by choosing one vertex from each triangle.
Constitutive relation — Any neighbor to a vertex in the independent set S cannot be in S because these vertices are disjoint by the independent set definition.
Operating condition — That is, it is a set S such that every pair of vertices in S is connected by an edge and every vertex not in S is missing an edge to at least one vertex in S.
Recognition evidence — The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence.
Admissible variation — Choose a random set of vertices S ⊆ V, by selecting each vertex v independently with probability 1/(2d(v)), where d is the degree of v (the number of neighbours of v).
Characteristic consequence — PROOF: Build a directed version of G by directing each edge to the node with the higher degree (breaking ties arbitrarily).
Failure boundary — So the number of good edges drops by at least a constant factor each step.

What It Is Not¶

Not the whole field of computer science and information systems. The node requires the specific identity stated by In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set.
Not an over-broad reading. However, it is not true that every edge of the graph has at least one, or even one endpoint in S.
Not an over-broad reading. As a result, every edge of the graph has at least one endpoint not in S .
Not an over-broad reading. A maximal clique is a set of vertices that induces a complete subgraph, and that is not a subset of the vertices of any larger complete subgraph.
Not automatically Independent set (graph theory). Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Maximal independent set applies literally inside computer science and information systems wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. The phrase "maximal independent set" is also used to describe maximal subsets of independent elements in mathematical structures other than graphs, and in particular in vector spaces and matroids.
Definition. For a graph G = (V, E) , an independent set S is a maximal independent set if for v \in V , one of the following is true.
Definition. N(v) \cap S \neq \emptyset where N(v) denotes the neighbors of v.
Definition. The above can be restated as a vertex either belongs to the independent set or has at least one neighbor vertex that belongs to the independent set.
Definition. As a result, every edge of the graph has at least one endpoint not in S .
Definition. However, it is not true that every edge of the graph has at least one, or even one endpoint in S.

Outside computer science and information systems, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Maximal independent set names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. The strongest recognition evidence in the frozen account is: The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However, it is not true that every edge of the graph has at least one, or even one endpoint in S. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Maximal independent set compresses multiple computer science and information systems details into a stable diagnostic relation. The source shows both the central mechanism—any neighbor to a vertex in the independent set S cannot be in S because these vertices are disjoint by the independent set definition.—and the practical consequence—pROOF: Build a directed version of G by directing each edge to the node with the higher degree (breaking ties arbitrarily). This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the computer science and information systems entities to which the claim applies.
State the relation. Use the source-grounded identity: In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set.
Check operation and conditions. That is, it is a set S such that every pair of vertices in S is connected by an edge and every vertex not in S is missing an edge to at least one vertex in S.
Demand recognition evidence. The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence.
Test variation. Change an implementation or setting while preserving choose a random set of vertices S ⊆ V, by selecting each vertex v independently with probability 1/(2d(v)), where d is the degree of v (the number of neighbours of v).
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Maximal independent set transfers literally when a new case preserves the same carrier type, relation, and recognition test. The phrase "maximal independent set" is also used to describe maximal subsets of independent elements in mathematical structures other than graphs, and in particular in vector spaces and matroids. For a graph G = (V, E) , an independent set S is a maximal independent set if for v \in V , one of the following is true.

Beyond the home domain. No canonical parent is asserted for Maximal independent set. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Break ties arbitrarily, e.g. using a lexicographic order on the vertex names. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set; recognition evidence → The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence

Applied / In Practice¶

A worst-case graph, in which the average number of steps is \Theta(\log(n)) , is a graph made of n/2 connected components, each with 2 nodes. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Finding a single maximal independent setSequential algo; invariant → In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set; boundary → the case exits the class when however, it is not true that every edge of the graph has at least one, or even one endpoint in S

Structural Tensions¶

T1 — Stable identity versus admissible variation. However, it is not true that every edge of the graph has at least one, or even one endpoint in S. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. As a result, every edge of the graph has at least one endpoint not in S . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. A maximal clique is a set of vertices that induces a complete subgraph, and that is not a subset of the vertices of any larger complete subgraph. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. That is, it is a set S such that every pair of vertices in S is connected by an edge and every vertex not in S is missing an edge to at least one vertex in S. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Any maximal independent set in this graph is formed by choosing one vertex from each triangle. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Maximal independent set literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. Any neighbor to a vertex in the independent set S cannot be in S because these vertices are disjoint by the independent set definition. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Maximal independent set distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Maximal independent set is structural-leaning. Its structural side is the repeatable organization summarized by In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. Its framed side is the computer science and information systems vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: That is, it is a set S such that every pair of vertices in S is connected by an edge and every vertex not in S is missing an edge to at least one vertex in S. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Any maximal independent set in this graph is formed by choosing one vertex from each triangle. Any neighbor to a vertex in the independent set S cannot be in S because these vertices are disjoint by the independent set definition. It further constrains recognition and variation through: That is, it is a set S such that every pair of vertices in S is connected by an edge and every vertex not in S is missing an edge to at least one vertex in S. The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence.

What is domain-bound. computer science and information systems supplies the operative entities, technical vocabulary, warrants, and exceptions that make Maximal independent set literal. Its documented scope includes the condition that The phrase "maximal independent set" is also used to describe maximal subsets of independent elements in mathematical structures other than graphs, and in particular in vector spaces and matroids. Another bounded application condition is that For a graph G = (V, E) , an independent set S is a maximal independent set if for v \in V , one of the following is true. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Choose a random set of vertices S ⊆ V, by selecting each vertex v independently with probability 1/(2d(v)), where d is the degree of v (the number of neighbours of v).—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Maximal independent set. The reviewed identity is: In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Maximal independent set sits in a crowded region of the domain-specific corpus (36^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Data Structures & Graph Variants (17 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set?
Independent set (graph theory). A vertex subset of a graph in which no two selected vertices are adjacent. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Matching (graph theory). A set of graph edges with no shared endpoint. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Dissociation number. The maximum number of vertices in a graph whose induced subgraph has maximum degree at most one. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Maximal independent set remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside computer science and information systems lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Maximal_independent_set (revision 1351521773).
Preserved source candidate: http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_749.html
Preserved source candidate: https://web.archive.org/web/20070709192029/http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_749.html
Preserved source candidate: http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_750.html
Preserved source candidate: https://web.archive.org/web/20070708145819/http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_750.html
Preserved source candidate: http://www.cc.gatech.edu/~vigoda/RandAlgs/MIS.pdf
Preserved source candidate: http://dcg.ethz.ch/lectures/podc_allstars/lecture/chapter7.pdf
Preserved source candidate: https://web.archive.org/web/20150221232848/http://dcg.ethz.ch/lectures/podc_allstars/lecture/chapter7.pdf
Preserved source candidate: http://epubs.siam.org/doi/10.1137/0212053

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.