[Paper Review] Generating All Maximal Induced Subgraphs for Hereditary, Connected-Hereditary and Rooted-Hereditary Properties

Sara Cohen, Yehoshua Sagiv|ArXiv.org|Oct 17, 2004

Advanced Graph Theory Research15 references3 citations

TL;DR

This paper presents a general framework for efficiently generating all maximal induced subgraphs with hereditary, connected-hereditary, or rooted-hereditary properties in a given graph. By reducing the general problem to restricted subproblems—such as those with a fixed root or vertex constraint—it enables efficient solutions using incremental polynomial time, total polynomial time, or P-enumerable complexity, significantly simplifying the design of algorithms for properties like cliques, independent sets, and bipartite subgraphs.

ABSTRACT

The problem of computing all maximal induced subgraphs of a graph G that have a graph property P, also called the maximal P-subgraphs problem, is considered. This problem is studied for hereditary, connected-hereditary and rooted-hereditary graph properties. The maximal P-subgraphs problem is reduced to restricted versions of this problem by providing algorithms that solve the general problem, assuming that an algorithm for a restricted version is given. The complexity of the algorithms are analyzed in terms of total polynomial time, incremental polynomial time and the complexity class P-enumerable. The general results presented allow simple proofs that the maximal P-subgraphs problem can be solved efficiently (in terms of the input and output) for many different properties.

Motivation & Objective

• To address the computational challenge of generating all maximal induced subgraphs with a given graph property P in a large graph.
• To reduce the general maximal P-subgraphs problem to simpler, restricted versions that are easier to solve algorithmically.
• To establish conditions under which the general problem is solvable in total polynomial time, incremental polynomial time (PINC), or in the complexity class P-enumerable.
• To provide a unified framework applicable to well-known graph properties such as cliques, independent sets, and bipartite graphs.
• To enable practical applications in databases, particularly for computing maximal query answers under incomplete information semantics.

Proposed method

• Reduce the general maximal P-subgraphs problem to input-restricted or vertex-restricted variants, such as computing maximal P-subgraphs containing a fixed vertex or root.
• Use a recursive algorithm, GenWithVertex⟨P⟩, that builds maximal induced subgraphs incrementally by adding vertices while maintaining the property P.
• Adapt the algorithm GenHered⟨P⟩ to handle rooted-hereditary and connected-hereditary properties by carefully ordering vertex processing to ensure incremental output.
• Leverage the fact that if the restricted problem is solvable in polynomial time, then the general problem inherits complexity guarantees such as PINC or P-enumerable status.
• Introduce a stack-based mechanism (Stack2) to track and output new maximal subgraphs during recursion, enabling incremental output generation.
• Apply the framework to derive complexity results for specific properties, such as bipartite and star graphs, by reducing them to simpler restricted cases.

Experimental results

Research questions

• RQ1 Under what conditions can the general problem of generating all maximal induced subgraphs with a hereditary property P be solved efficiently?
• RQ2 How can the maximal P-subgraphs problem be reduced to simpler, restricted subproblems that are easier to solve?
• RQ3 What complexity guarantees (e.g., total polynomial time, incremental polynomial time, P-enumerable) can be inherited from restricted versions of the problem?
• RQ4 Can the framework be applied to practical problems such as query answering in databases with incomplete information?
• RQ5 For which classes of graph properties—hereditary, connected-hereditary, or rooted-hereditary—is the maximal P-subgraphs problem guaranteed to be in PINC or P-enumerable?

Key findings

• The maximal P-subgraphs problem is P-enumerable for rooted-hereditary properties when the input graph is acyclic, and for connected-hereditary properties when the underlying undirected graph is a tree.
• The maximal P-subgraphs problem is solvable in incremental polynomial time (PINC) if the input-restricted version of the problem is solvable in polynomial time.
• The maximal "bipartite"-subgraph problem is solvable in total polynomial time, improving upon prior results.
• The framework enables efficient computation of maximal subgraphs for properties like independent sets, star graphs, and cliques by reducing them to simpler restricted subproblems.
• The approach allows for returning the first k maximal induced subgraphs in polynomial time for any constant k, under the same complexity assumptions.
• The results extend to database applications, showing that semantics like full disjunctions in incomplete data can be modeled as rooted-hereditary properties and computed efficiently.