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Chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see holes in graph theory). Every chordal graph is a strangulated graph , a graph in which every peripheral cycle is a triangle, because peripheral cycles are a special case of induced cycles.
Every graph is the line graph of some hypergraph, but, given a fixed edge size k, not every graph is a line graph of some k-uniform hypergraph. A main problem is to characterize those that are, for each k ≥ 3. A hypergraph is linear if each pair of hyperedges intersects in at most one vertex. Every graph is the line graph, not only of some ...
A triangle-free graph is a graph with no induced cycle of length three. The cographs are exactly the graphs with no induced path of length three. The chordal graphs are the graphs with no induced cycle of length four or more. The even-hole-free graphs are the graphs containing no induced cycles with an even number of vertices.
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
The perfect graphs are exactly the graphs for which this function is the identity, both for the graph itself and for all its induced subgraphs. [59] The equality of the clique number and chromatic number in perfect graphs has motivated the definition of other graph classes, in which other graph invariants are set equal to each other.
Every -perfect graph must be an even-hole-free graph, because even cycles have chromatic number two and degeneracy two, not matching the equality in the definition of -perfect graphs. If a graph and its complement graph are both even-hole-free, they are both β {\displaystyle \beta } -perfect.
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality.
The graph of a continuous piecewise linear function on a compact interval is a polygonal chain. (*) A linear function satisfies by definition f ( λ x ) = λ f ( x ) {\displaystyle f(\lambda x)=\lambda f(x)} and therefore in particular f ( 0 ) = 0 {\displaystyle f(0)=0} ; functions whose graph is a straight line are affine rather than linear .