Feasible Graphs in Extremal Graph Theory

• Feasible Graphs are defined as families that can realize every possible edge count for every graph order, ensuring a full combinatorial spectrum.
• They are characterized by induced G-free conditions, with feasibility failing only for specific graphs such as Kₖ, Kₖ\K₂, and their complements for k ≥ 2.
• The paper introduces constructive methods like the Universal Elimination Process and {K₃,K₂}-elimination to systematically achieve every edge count while preserving induced-freeness.

In extremal graph theory, a family of graphs $\mathcal F$ is called feasible if every admissible order–size pair is realized inside the family: for every $n\ge 1$ and every $0\le m\le \binom{n}{2}$, there exists a graph $G\in\mathcal F$ with exactly $n$ vertices and exactly $m$ edges. For families defined by forbidding a fixed induced subgraph, the feasibility problem has a complete answer: for a graph $G$, the family $\mathcal F(G)$ of all induced $G$-free graphs is feasible if and only if $G$ is not isomorphic to $n\ge 1$0, $n\ge 1$1, $n\ge 1$2, or $n\ge 1$3 for any $n\ge 1$4 (Caro et al., 2023).

1. Definition of feasibility and the basic formalism

The feasibility problem asks whether a graph family realizes all possible edge counts at every order. If $n\ge 1$5 is an infinite family of graphs, feasibility means that for each fixed $n\ge 1$6, the family contains graphs with every edge count from $n\ge 1$7 to $n\ge 1$8 (Caro et al., 2023).

A convenient encoding uses the sets

$n\ge 1$9

and

$0\le m\le \binom{n}{2}$0

Under this notation, $0\le m\le \binom{n}{2}$1 is feasible exactly when $0\le m\le \binom{n}{2}$2 is empty (Caro et al., 2023).

This formulation is unusually broad. It does not optimize a single statistic, such as an extremal edge count, and it does not privilege dense or sparse regimes. Instead, it asks whether the family is combinatorially rich enough to interpolate the full discrete range of possible sizes at each order. This makes the notion especially suited to families defined by hereditary constraints, and in particular to families defined by forbidding one fixed graph as an induced subgraph.

2. Induced $0\le m\le \binom{n}{2}$3-free families

For a fixed graph $0\le m\le \binom{n}{2}$4, the notation $0\le m\le \binom{n}{2}$5 denotes the family of all graphs that are induced $0\le m\le \binom{n}{2}$6-free, meaning that they contain no induced copy of $0\le m\le \binom{n}{2}$7. Here the induced condition is essential: a set of vertices induces a copy of $0\le m\le \binom{n}{2}$8 only if both adjacency and non-adjacency match exactly, which is stricter than merely containing $0\le m\le \binom{n}{2}$9 as a subgraph (Caro et al., 2023).

Complements play a central role. If $G\in\mathcal F$0 denotes the complement of $G\in\mathcal F$1, then induced $G\in\mathcal F$2-freeness is equivalent to induced $G\in\mathcal F$3-freeness under complementation: $G\in\mathcal F$4 This symmetry, stated as Proposition 2, immediately reduces many cases to complementary pairs and explains why the final classification is closed under complementation (Caro et al., 2023).

The induced setting is also where the feasibility problem becomes structurally sharp. Ordinary subgraph exclusion often enforces monotone density restrictions, whereas induced exclusion can constrain both sparse and dense configurations. The resulting question is therefore not whether a family is large in cardinality, but whether it is sufficiently flexible across the entire $G\in\mathcal F$5-grid.

3. Complete characterization of feasible induced-free families

The main theorem gives an exact characterization: $G\in\mathcal F$6 Equivalently, the only obstructions to feasibility are the four graph types

$G\in\mathcal F$7

(Caro et al., 2023).

The excluded types are as follows.

Type	Description
$G\in\mathcal F$8	the complete graph on $G\in\mathcal F$9 vertices
$n$0	the graph obtained from $n$1 by deleting one edge
$n$2	the edgeless graph on $n$3 vertices
$n$4	the complement of $n$5

Since $n$6 is a clique missing one edge, its complement has exactly one edge and $n$7 isolated vertices: $n$8 Thus the non-feasible induced-forbidden patterns are precisely a clique, a near-clique, an independent set, and a single edge plus isolated vertices, together with their complementary forms (Caro et al., 2023).

The theorem is exact in both directions. If $n$9 is one of these forms, induced $m$0-freeness is too restrictive to realize all edge counts. If $m$1 is not one of these forms, then for every $m$2 and every $m$3 with $m$4, there exists an induced $m$5-free graph on $m$6 vertices with exactly $m$7 edges (Caro et al., 2023).

4. Proof architecture and structural mechanisms

The proof is graph-theoretic and is organized around explicit constructions that realize prescribed edge counts while preserving induced-freeness. One central device is the Universal Elimination Process (UEP), which starts from $m$8 and deletes edges systematically, first isolating $m$9, then $G$0, and so on, until the empty graph is reached. Along this process, graphs with every possible number of edges appear (Caro et al., 2023).

The maximal induced subgraphs produced by UEP have the form

$G$1

with $G$2 and $G$3. This construction immediately yields feasibility for several induced-free families, including $G$4-free graphs for $G$5, $G$6-free graphs for $G$7, and $G$8-free graphs for $G$9. It also shows that certain split-like graphs $\mathcal F(G)$0 are feasible unless they fall into the trivial forbidden cases (Caro et al., 2023).

A second method, called $\mathcal F(G)$1-elimination, is introduced for harder configurations, especially graphs close to $\mathcal F(G)$2 or related split graphs. Its key lemma states that for $\mathcal F(G)$3 and $\mathcal F(G)$4, there exist integers $\mathcal F(G)$5 such that

$\mathcal F(G)$6

and $\mathcal F(G)$7 is a subgraph of $\mathcal F(G)$8. This allows one to delete exactly $\mathcal F(G)$9 edges from a clique in controlled pieces while avoiding the relevant induced forbidden graph (Caro et al., 2023).

The case analysis reduces general $G$0 to the structured family

$G$1

From there, the proof separates trivially non-feasible cases from cases handled by UEP or $G$2-elimination, and then checks the remaining structured instances individually. A particularly important observation is that when $G$3 and $G$4, the complement of $G$5 with $G$6 contains an induced claw, so any claw-free graph is automatically $G$7-free; since claw-free graphs are feasible, this implies feasibility of those $G$8-free families (Caro et al., 2023).

5. Comparison with other graph families

The feasibility problem can behave very differently for other natural graph classes. For the family of line graphs, the answer is negative: not every pair $G$9 is realizable. The paper on line graphs defines $G$0 as feasible if there exists a graph $G$1 such that $G$2 and $G$3, and proves that for fixed $G$4, the non-feasible values of $G$5 form disjoint blocks of consecutive integers which are completely determined (Caro et al., 2021).

The smallest classical obstruction is $G$6, which is realized only by $G$7, not a line graph. More generally, for fixed $G$8, the non-feasible values are exactly the integers in

$G$9

for the stated range of $n\ge 1$00 (Caro et al., 2021). In this sense, line graphs are sharply non-feasible even though they form a large and classical hereditary class.

By contrast, the family of claw-free graphs is feasible: for every $n\ge 1$01 and every $n\ge 1$02, there exists a claw-free graph on $n\ge 1$03 vertices and $n\ge 1$04 edges (Caro et al., 2021). This contrast is instructive because line graphs are claw-free, yet the larger claw-free family fills the entire order–size grid while the more rigid subclass of line graphs leaves explicit gaps. A plausible implication is that feasibility is sensitive not merely to forbidden induced subgraphs, but to how strongly a structural representation constrains degree concentration and local overlap patterns.

6. Related feasible-region notions in combinatorics

The word feasible also appears in several adjacent combinatorial frameworks, where it denotes a realizable region of limiting statistics rather than a realizable order–size pair. For induced subgraph densities, the feasible region $n\ge 1$05 of a graph $n\ge 1$06 is the set of all limit points $n\ge 1$07 such that there exists a sequence of graphs whose edge densities approach $n\ge 1$08 and whose induced $n\ge 1$09-densities approach $n\ge 1$10. This region is always of the form

$n\ge 1$11

and the boundary functions are continuous and almost everywhere differentiable (Liu et al., 2021).

For consecutive permutation patterns, the feasible region $n\ge 1$12 consists of all possible limiting vectors of consecutive pattern densities. It is identified exactly with the cycle polytope of the overlap graph $n\ge 1$13: $n\ge 1$14 This description yields the defining equations, the dimension $n\ge 1$15, the vertices, and the face structure (Borga et al., 2019).

For tournament profiles, the feasible region is the set of density vectors of tournaments with at most $n\ge 1$16 vertices arising from tournamentons. Its dimension is not governed merely by strongly connected tournaments; rather, for every $n\ge 1$17, it is equal to the number of non-trivial Lyndon tournaments with at most $n\ge 1$18 vertices (Kral et al., 2023).

These related usages do not coincide with feasibility of a graph family in the order–size sense. They nonetheless reflect the same organizing principle: a combinatorial class is studied through the set of all statistics it can realize, and the main structural question becomes the exact shape of that realizability domain. Within that broader landscape, the feasibility classification for induced $n\ge 1$19-free graphs stands out for giving a complete obstruction set with only four graph types (Caro et al., 2023).