---
title: Feasible Graphs in Extremal Graph Theory
url: https://www.emergentmind.com/topics/feasible-graph
type: topic
---

# Feasible Graphs in Extremal Graph Theory

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 \(K_k\), \(K_k\backslash K_2\), \(\overline{K_k}\), or \(\overline{K_k\backslash K_2}\) for any \(k\ge 2\) [2311.01082].

## 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 \(\mathcal F\) is an infinite family of graphs, feasibility means that for each fixed \(n\), the family contains graphs with every edge count from \(0\) to \(\binom{n}{2}\) [2311.01082].

A convenient encoding uses the sets
\[
FP(\mathcal F)=\{(n,m): \text{there is a graph }G\in\mathcal F\text{ with }|V(G)|=n,\ e(G)=m\},
\]
and
\[
\overline{FP}(\mathcal F)=\{(n,m): \text{no graph in }\mathcal F\text{ has }n\text{ vertices and }m\text{ edges}\}.
\]
Under this notation, \(\mathcal F\) is feasible exactly when \(\overline{FP}(\mathcal F)\) is empty [2311.01082].

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 \(G\)-free families

For a fixed graph \(G\), the notation \(\mathcal F(G)\) denotes the family of all graphs that are **induced \(G\)-free**, meaning that they contain no induced copy of \(G\). Here the induced condition is essential: a set of vertices induces a copy of \(G\) only if both adjacency and non-adjacency match exactly, which is stricter than merely containing \(G\) as a subgraph [2311.01082].

Complements play a central role. If \(\overline G\) denotes the complement of \(G\), then induced \(G\)-freeness is equivalent to induced \(\overline G\)-freeness under complementation:
\[
\mathcal F(G)\text{ is feasible } \Longleftrightarrow \mathcal F(\overline G)\text{ is feasible}.
\]
This symmetry, stated as Proposition 2, immediately reduces many cases to complementary pairs and explains why the final classification is closed under complementation [2311.01082].

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 \((n,m)\)-grid.

## 3. Complete characterization of feasible induced-free families

The main theorem gives an exact characterization:
\[
\mathcal F(G)\text{ is feasible } \Longleftrightarrow  G\notin \{K_k,\ K_k\backslash K_2,\ \overline{K_k},\ \overline{K_k\backslash K_2}\}\quad (k\ge 2).
\]
Equivalently, the only obstructions to feasibility are the four graph types
\[
\mathrm{TNF}=\{K_k,\ K_k\backslash K_2,\ \overline{K_k},\ \overline{K_k\backslash K_2}\},\qquad k\ge 2
\]
[2311.01082].

The excluded types are as follows.

| Type | Description |
|---|---|
| \(K_k\) | the complete graph on \(k\) vertices |
| \(K_k\backslash K_2\) | the graph obtained from \(K_k\) by deleting one edge |
| \(\overline{K_k}\) | the edgeless graph on \(k\) vertices |
| \(\overline{K_k\backslash K_2}\) | the complement of \(K_k\backslash K_2\) |

Since \(K_k\backslash K_2\) is a clique missing one edge, its complement has exactly one edge and \(k-2\) isolated vertices:
\[
\overline{K_k\backslash K_2}=K_2\cup (k-2)K_1.
\]
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 [2311.01082].

The theorem is exact in both directions. If \(G\) is one of these forms, induced \(G\)-freeness is too restrictive to realize all edge counts. If \(G\) is not one of these forms, then for every \(n\) and every \(m\) with \(0\le m\le \binom{n}{2}\), there exists an induced \(G\)-free graph on \(n\) vertices with exactly \(m\) edges [2311.01082].

## 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 \(K_n\) and deletes edges systematically, first isolating \(v_1\), then \(v_2\), and so on, until the empty graph is reached. Along this process, graphs with every possible number of edges appear [2311.01082].

The maximal induced subgraphs produced by UEP have the form
\[
H(p,q,r)= (K_p \backslash K_{1,q}) \cup rK_1,
\]
with \(p-1\ge q\ge 0\) and \(p+r=n\). This construction immediately yields feasibility for several induced-free families, including \(K_{1,r}\)-free graphs for \(r\ge 3\), \(P_r\)-free graphs for \(r\ge 3\), and \(rK_2\)-free graphs for \(r\ge 2\). It also shows that certain split-like graphs \(S(p,r)=K_p+\overline{K_r}\) are feasible unless they fall into the trivial forbidden cases [2311.01082].

A second method, called **\(\{K_3,K_2\}\)-elimination**, is introduced for harder configurations, especially graphs close to \(K_4\backslash K_2\) or related split graphs. Its key lemma states that for \(n\ge 2\) and \(0\le t\le n-2\), there exist integers \(x,y\ge 0\) such that
\[
3x+y=t
\]
and \(xK_3\cup yK_2\) is a subgraph of \(K_n\). This allows one to delete exactly \(t\) edges from a clique in controlled pieces while avoiding the relevant induced forbidden graph [2311.01082].

The case analysis reduces general \(G\) to the structured family
\[
H(p,q,r)= (K_p \backslash K_{1,q}) \cup rK_1.
\]
From there, the proof separates trivially non-feasible cases from cases handled by UEP or \(\{K_3,K_2\}\)-elimination, and then checks the remaining structured instances individually. A particularly important observation is that when \(p\ge 4\) and \(q=1\), the complement of \(H(p,1,r)\) with \(r\ge 1\) contains an induced claw, so any claw-free graph is automatically \(H(p,1,r)\)-free; since claw-free graphs are feasible, this implies feasibility of those \(H(p,1,r)\)-free families [2311.01082].

## 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 \((n,m)\) is realizable. The paper on line graphs defines \((N,M)\) as feasible if there exists a graph \(G\) such that \(e(G)=N\) and \(e(L(G))=M\), and proves that for fixed \(N\ge 5\), the non-feasible values of \(M\) form disjoint blocks of consecutive integers which are completely determined [2107.13806].

The smallest classical obstruction is \((N,M)=(5,9)\), which is realized only by \(K_5\setminus\{e\}\), not a line graph. More generally, for fixed \(N\ge 5\), the non-feasible values are exactly the integers in
\[
\left[\binom{N-t}{2}+\binom{t+2}{2},\ \binom{N-t+1}{2}-1\right]
\]
for the stated range of \(t\) [2107.13806]. 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 \(n\) and every \(0\le m\le \binom{n}{2}\), there exists a claw-free graph on \(n\) vertices and \(m\) edges [2107.13806]. 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 \(\Omega_{\rm ind}(F)\) of a graph \(F\) is the set of all limit points \((x,y)\in[0,1]^2\) such that there exists a sequence of graphs whose edge densities approach \(x\) and whose induced \(F\)-densities approach \(y\). This region is always of the form
\[
\Omega_{\rm ind}(F)=\{(x,y)\in[0,1]\times \mathbb R:\ i(F,x)\le y\le I(F,x)\},
\]
and the boundary functions are continuous and almost everywhere differentiable [2106.16203].

For consecutive permutation patterns, the feasible region \(P_k\) consists of all possible limiting vectors of consecutive pattern densities. It is identified exactly with the cycle polytope of the overlap graph \(\ValGraph[k]\):
\[
P_k=P(\ValGraph[k]).
\]
This description yields the defining equations, the dimension \(\dim P_k=k!-(k-1)!\), the vertices, and the face structure [1910.02233].

For tournament profiles, the feasible region is the set of density vectors of tournaments with at most \(k\) vertices arising from tournamentons. Its dimension is not governed merely by strongly connected tournaments; rather, for every \(k\ge 3\), it is equal to the number of non-trivial Lyndon tournaments with at most \(k\) vertices [2310.19482].

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 \(G\)-free graphs stands out for giving a complete obstruction set with only four graph types [2311.01082].

Source: https://www.emergentmind.com/topics/feasible-graph