Extended Edge Elimination (EEE)

• Extended Edge Elimination is a graph invariant defined recursively by deletion, contraction, and extraction, generalizing Tutte, matching, and chromatic polynomials.
• EEE is applied in combinatorial optimization to preprocess symmetric TSP instances by eliminating edges absent from every optimum tour using 2-opt, 3-opt, and potential-point geometry techniques.
• The framework supports reconstructibility and complexity analysis, linking thickening identities with hardness proofs and extraction of key structural graph parameters.

Extended Edge Elimination (EEE) denotes two distinct edge-centered constructions that appear in the literature under closely related names. In graph polynomial theory, the extended edge-elimination polynomial, written as $\xi(G;x,y,z)$ or $E(G;x,y,z)$, is the unique graph invariant defined by deletion, contraction, and extraction recurrences; it is described as the most general linear deletion–contraction–extraction polynomial and has specializations including Tutte-type, matching, covered-components, and bivariate chromatic polynomials (0801.1600). In combinatorial optimization, Extended Edge Elimination is a preprocessing method for symmetric TSP instances that removes edges provably absent from every optimum tour by combining 2-opt incompatibility, 3-opt arguments, potential-point geometry, close-point rules, and depth-limited backtracking (Hougardy et al., 2014).

1. Recurrence framework for the graph polynomial

In the graph-polynomial literature, the object is defined on finite undirected graphs, with loops and multiple edges allowed in the universal formulation. The three edge operations are deletion, contraction, and extraction. For an edge $e=uv$, deletion produces $G-e$, contraction produces $G/e$, and extraction produces the graph obtained by deleting $e$, deleting both endpoints, and deleting all incident edges. Different sources write extraction as $G\Uparrow e$, $G_{\dagger e}$, or $G_{(e)}$; these are notational variants of the same elimination pattern (Trinks, 2012).

The recurrence is

$$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$

Equivalent formulations use $E(G;x,y,z)$0 in place of $E(G;x,y,z)$1 and add multiplicativity on disjoint unions:

$E(G;x,y,z)$2

together with the base value $E(G;x,y,z)$3 (Trinks, 2012).

The base cases may also be stated as

$E(G;x,y,z)$4

For paths, this becomes $E(G;x,y,z)$5 and $E(G;x,y,z)$6 (Dohmen, 2014). The supplied sources state that these axioms determine the invariant uniquely and that the result is independent of the choice of eliminated edge. A common source of ambiguity is terminological rather than mathematical: the acronym “EEE” is used both for this polynomial and for the TSP preprocessing method discussed below.

2. Equivalent expansions and principal specializations

Two global expansions are emphasized for the polynomial formulation. The first is a subgraph-enumeration expansion via

$E(G;x,y,z)$7

where the sum ranges over vertex-edge-subgraphs and $E(G;x,y,z)$8 is the number of connected components. The exact change of variables recorded in the literature is

$E(G;x,y,z)$9

and

$e=uv$0

Substituting into the vertex-edge expansion of $e=uv$1 yields

$e=uv$2

The second is a coloring-enumeration expansion through the trivariate chromatic polynomial $e=uv$3, which also satisfies a recurrence of type $e=uv$4–$e=uv$5 and is linked to $e=uv$6 by explicit substitutions (Trinks, 2012).

Several specializations are central. Setting $e=uv$7 gives

$e=uv$8

the generating function of spanning subgraphs by number of components and edges. The ordinary Tutte polynomial is then recovered by the stated change of variables

$e=uv$9

The bivariate chromatic polynomial of Dohmen–Pönitz–Tittmann is obtained as

$G-e$0

The bivariate matching polynomial is

$G-e$1

and the covered-components polynomial is

$G-e$2

These substitutions show why the polynomial is presented as a simultaneous generalization of several classical graph polynomials (0801.1600).

This also suggests an interpretive division of labor among the variables: the exposition of the universal edge-elimination polynomial states that $G-e$3 tracks surviving isolated vertices, $G-e$4 tracks contractions of edges, and $G-e$5 tracks deletions of edges plus their endpoints (Dohmen, 2014).

3. Closed forms for paths and cycles

For the $G-e$6-vertex path $G-e$7, the recurrence specializes to a second-order homogeneous linear recurrence:

$G-e$8

where $G-e$9. Its characteristic equation is

$G/e$0

with discriminant

$G/e$1

Writing

$G/e$2

the closed form for $G/e$3 is

$G/e$4

The same source also gives the $G/e$5 trigonometric form and the double-root case $G/e$6 (Dohmen, 2014).

A combinatorial binomial-sum expansion is recorded as

$G/e$7

For trees, a simpler formula holds: for an $G/e$8-vertex tree,

$G/e$9

The tree formula is derived in the thickening paper by induction using the fact that every edge of a tree is a bridge (0801.1600).

For the cycle $e$0, the recurrence becomes

$e$1

Substituting the path solution yields the compact expression

$e$2

Equivalently, for $e$3,

$e$4

The cycle inherits the two geometric terms from the path, but the literature explicitly identifies the extra term $e$5 as a non-homogeneous “closing-term” created by the last edge on the cycle (Dohmen, 2014).

Ordinary generating functions are also known:

$e$6

and

$e$7

4. Thickening of edges and complexity of evaluation

A major structural identity concerns $e$8-thickening. If $e$9 is obtained from $G\Uparrow e$0 by replacing each edge by $G\Uparrow e$1 parallel copies, then in the auxiliary $G\Uparrow e$2-form one has

$G\Uparrow e$3

Translating back to the original variables gives the boxed identity

$G\Uparrow e$4

The proof proceeds through an auxiliary polynomial

$G\Uparrow e$5

followed by a local doubling-edge calculation showing that doubling an edge replaces $G\Uparrow e$6 by $G\Uparrow e$7 (0801.1600).

This thickening identity is then combined with a point-to-point reduction from the bivariate chromatic polynomial to obtain a broad hardness statement. The paper states that for every rational triple $G\Uparrow e$8 satisfying

• $G\Uparrow e$9,
• $G_{\dagger e}$0,
• $G_{\dagger e}$1,
• $G_{\dagger e}$2,

computing $G_{\dagger e}$3 from $G_{\dagger e}$4 is #P-hard. It is then summarized as intractability outside a certain two-dimensional exceptional set in $G_{\dagger e}$5 (0801.1600).

The same identity is already informative at the level of examples. For a single edge $G_{\dagger e}$6, thickening to a $G_{\dagger e}$7-fold multiedge reduces evaluation to

$G_{\dagger e}$8

which expresses the multiedge instance through an ordinary edge. This suggests why thickening is effective both as an algebraic transformation and as a complexity-theoretic reduction device.

5. Reconstructibility and encoded graph structure

The polynomial has nontrivial structural consequences beyond evaluation formulas. For simple graphs with at least three vertices, the extended edge-elimination polynomial is reconstructible from its polynomial deck:

$G_{\dagger e}$9

The reconstructibility theorem is stated as follows: if $G_{(e)}$0 is a simple graph with $G_{(e)}$1, then $G_{(e)}$2 is determined by the multiset $G_{(e)}$3 (Trinks, 2012).

The proof strategy uses the equivalent subgraph counting polynomial $G_{(e)}$4. Kelly’s lemma is applied to show reconstructibility of $G_{(e)}$5, and the invertible transformation between $G_{(e)}$6 and $G_{(e)}$7 then transfers the result. In this sense, the recurrence-defined invariant is not merely computable recursively; it can also be recovered from vertex-deleted data.

A second structural theorem extracts the degree sequence through the coloring form $G_{(e)}$8. If $G_{(e)}$9 denotes the number of vertices of degree $$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$0, then

$$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$1

where $$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$2. The proof sketch given in the source interprets $$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$3 so that exactly one vertex is colored from a palette of size $$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$4, all remaining vertices are forced to the single proper color, and the exponent of $$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$5 counts edges not incident to the uniquely colored vertex. The coefficient extraction therefore recovers the degree statistics (Trinks, 2012).

Worked examples confirm the recurrence and the summation forms. For the one-edge graph $$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$6,

$$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$7

For the 3-vertex path $$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$8,

$$\xi(G;x,y,z)=\xi(G-e;x,y,z)+y\,\xi(G/e;x,y,z)+z\,\xi(G\Uparrow e;x,y,z).$$9

The sources state that both the subgraph-sum expansion and the coloring-sum expansion reproduce the latter expression exactly.

6. Extended Edge Elimination for symmetric TSP instances

In the TSP literature, Extended Edge Elimination is not a graph polynomial but a preprocessing procedure on a complete graph TSP instance $E(G;x,y,z)$00. An edge is called useless if it belongs to no optimum tour. EEE returns a subset $E(G;x,y,z)$01 containing all optimum tours’ edges by repeatedly applying elimination rules that extend basic pairwise incompatibility to triples of edges, potential-point geometry, and shallow search (Hougardy et al., 2014).

The basic pairwise incompatibility rule is the familiar 2-opt test. If

$E(G;x,y,z)$02

then $E(G;x,y,z)$03 and $E(G;x,y,z)$04 cannot both lie in an optimum tour. The source also states the compatibility lemma in contraposed form:

$E(G;x,y,z)$05

The main extension is the Main Edge Elimination theorem. For an edge $E(G;x,y,z)$06, one seeks two potential points $E(G;x,y,z)$07 whose compatible neighborhoods are covered by two disjoint cones. If the stated length-difference inequalities hold,

$E(G;x,y,z)$08

and

$E(G;x,y,z)$09

then $E(G;x,y,z)$10 is useless. The proof sketch is explicitly 3-opt: in any tour containing $E(G;x,y,z)$11, the $E(G;x,y,z)$12- and $E(G;x,y,z)$13-neighbors must lie in opposite cones, so one of two specific 3-opt moves strictly shortens the tour (Hougardy et al., 2014).

Close-point elimination reduces the need for two potential points. Let

$E(G;x,y,z)$14

If for all $E(G;x,y,z)$15,

$E(G;x,y,z)$16

then $E(G;x,y,z)$17 is useless. Strong close-point elimination further uses a 2-opt+shortcut move and the metric excess $E(G;x,y,z)$18. If

$E(G;x,y,z)$19

then the triple $E(G;x,y,z)$20 is 3-incompatible (Hougardy et al., 2014).

For Euclidean instances, the paper gives a certification test for strongly potential points. Under $E(G;x,y,z)$21, with $E(G;x,y,z)$22 defined as half the integer-rounded nearest-neighbour distance,

$E(G;x,y,z)$23

With cones

$E(G;x,y,z)$24

the sufficient conditions are

$E(G;x,y,z)$25

where

$E(G;x,y,z)$26

and

$E(G;x,y,z)$27

If these hold, then in any optimum tour through $E(G;x,y,z)$28 the two $E(G;x,y,z)$29-neighbors cannot both lie in the same cone.

The algorithmic organization is a three-stage loop over edges: a fast main-theorem elimination using strongly potential points, close-point elimination on the resulting small candidate list, and depth-limited backtracking with maximum depth $E(G;x,y,z)$30. With a 2-d-tree and precomputed $E(G;x,y,z)$31-values, the main part of the algorithm is analyzed as $E(G;x,y,z)$32 for an $E(G;x,y,z)$33-vertex TSP instance (Hougardy et al., 2014).

The same source records both exact and empirical consequences. The abstract states that by combining the approach with Concorde, a large TSPLIB instance can be solved “more than 11 times faster than Concorde alone.” The detailed notes attribute this to Hougardy–Schroeder and state that TSPLIB instances of size up to $E(G;x,y,z)$34 saw an $E(G;x,y,z)$35 speed-up in total CPU time when Concorde was given only the pruned edge-set $E(G;x,y,z)$36. The paper also states that edges can be eliminated independently, so the method “parallelises perfectly on $E(G;x,y,z)$37 tasks” (Hougardy et al., 2014).

A common misconception is to treat the TSP method and the graph polynomial as variants of one framework. The supplied literature supports a narrower conclusion: they share the phrase “edge elimination,” but one is a recurrence-defined graph invariant and the other is a family of geometric and combinatorial pruning rules for optimum tours.