A framework for minimal hereditary classes of graphs of unbounded clique-width
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We create a framework for hereditary graph classes 𝒢^δ built on a two-dimensional grid of vertices and edge sets defined by a triple δ={α,β,γ} of objects that define edges between consecutive columns, edges between non-consecutive columns (called bonds), and edges within columns. This framework captures all previously proven minimal hereditary classes of graph of unbounded clique-width, and many new ones, although we do not claim this includes all such classes. We show that a graph class 𝒢^δ has unbounded clique-width if and only if a certain parameter 𝒩^δ is unbounded. We further show that 𝒢^δ is minimal of unbounded clique-width (and, indeed, minimal of unbounded linear clique-width) if another parameter ℳ^β is bounded, and also δ has defined recurrence characteristics. Both the parameters 𝒩^δ and ℳ^β are properties of a triple δ=(α,β,γ), and measure the number of distinct neighbourhoods in certain auxiliary graphs. Throughout our work, we introduce new methods to the study of clique-width, including the use of Ramsey theory in arguments related to unboundedness, and explicit (linear) clique-width expressions for subclasses of minimal classes of unbounded clique-width.
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