[Paper Review] Minimal forbidden induced subgraphs of graphs of bounded clique-width and bounded linear clique-width

Daniel Meister, Udi Rotics|arXiv (Cornell University)|Jun 10, 2013

Advanced Graph Theory Research1 references3 citations

TL;DR

This paper identifies minimal forbidden induced subgraphs for graphs of bounded clique-width and bounded linear clique-width by analyzing path power-based constructions and their subgraphs. It proves that the graphs $ Z_k $ and $ S_k^+ $ are minimal with respect to clique-width $ k+2 $, and their proper induced subgraphs have strictly smaller linear clique-width, establishing complete, minimal sets of forbidden subgraphs for bounded clique-width classes.

ABSTRACT

In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs of bounded clique-width and for graphs of bounded linear clique-width for arbitrary bounds.

Motivation & Objective

• To determine complete, minimal sets of forbidden induced subgraphs for graphs of bounded clique-width and bounded linear clique-width.
• To show that certain path power-based graphs, such as $ Z_k $ and $ S_k^+ $, are minimal in their clique-width and linear clique-width classes.
• To prove that all proper induced subgraphs of $ Z_k $ have linear clique-width at most $ k+1 $, establishing minimality of $ Z_k $ for clique-width $ k+2 $.
• To extend the characterization to arbitrary bounds on clique-width and linear clique-width using embedding techniques into larger graphs like $ J_k - z_g $.

Proposed method

• Constructs graphs $ Z_k $ as $ k $-path powers on $ k(k+1)+2 $ vertices with a specific vertex ordering to achieve high clique-width.
• Uses embedding arguments: shows that every proper induced subgraph of $ Z_k $ is isomorphic to an induced subgraph of $ J_k - z_g $, a graph with known linear clique-width at most $ k+1 $.
• Applies results from prior work on open $ k $-models and short-end $ k $-models to construct linear clique-width expressions with inactive labels.
• Employs vertex relabeling and recursive construction of linear $ (k+1) $-expressions for subgraphs of $ F_k $, the core structure of $ S_k^+ $, by partitioning the vertex set based on deletion position.
• Distinguishes cases based on the position of the deleted vertex $ x $, using deep and shallow rectangles in bubble model embeddings to ensure valid linear clique-width expressions.
• Uses automorphism and symmetry arguments to reduce the number of cases to consider, focusing on $ x $ in the first half of the vertex ordering.

Experimental results

Research questions

• RQ1 What are the minimal induced subgraphs that forbid a graph from having clique-width at most $ k $, for any $ k \geq 0 $?
• RQ2 How can one prove that a graph is minimal with respect to a given clique-width bound?
• RQ3 Can the forbidden subgraph characterization for bounded clique-width be extended to bounded linear clique-width?
• RQ4 What structural properties of path power-based graphs allow them to serve as minimal obstructions for bounded clique-width?
• RQ5 How do embedding techniques into larger graphs like $ J_k - z_g $ help in proving linear clique-width bounds for subgraphs?

Key findings

• The graph $ Z_k $ has clique-width exactly $ k+2 $, and all its proper induced subgraphs have linear clique-width at most $ k+1 $, proving its minimality for clique-width $ k+2 $.
• For $ k \geq 3 $, every proper induced subgraph of $ Z_k $ is isomorphic to an induced subgraph of $ J_k - z_g $, which has linear clique-width at most $ k+1 $.
• The graph $ S_k^+ $, constructed from a $ k $-path power with additional vertices, is minimal with respect to linear clique-width $ k+2 $, as all its proper induced subgraphs have linear clique-width at most $ k+1 $.
• The construction of linear $ (k+1) $-expressions for subgraphs of $ F_k $ (the core of $ S_k^+ $) is feasible regardless of the position of the deleted vertex, by using deep and shallow rectangle embeddings.
• For $ k=2 $, $ Z_2 $ is a path on 8 vertices; its subgraphs have linear clique-width at most 3, and a linear 3-expression can be explicitly constructed for $ Z_2 - v_4 $.
• The results imply that complete, minimal sets of forbidden induced subgraphs exist for graphs of bounded clique-width and bounded linear clique-width, for any bound $ k \geq 0 $.