Exact Unit-Ball Packing in Continuous Complete Multipartite Graphs

1. Introduction

The maximum independent-set problem is a central problem in graph theory, combinatorial optimization, and theoretical computer science. For a finite graph, an independent set is a subset of vertices containing no adjacent pair, and the corresponding optimization problem is one of Karp’s classical NP-complete problems [5]. In many geometric, network, and scheduling models, however, an edge is better regarded as a one-dimensional continuum rather than as a purely combinatorial adjacency. This leads to the study of metric graphs: geometric realizations of finite graphs in which edges are represented by intervals and distances are shortest-path distances in the resulting one-dimensional metric space [1,8].

The systematic study of continuous analogues of classical graph-optimization problems has recently been developed by Grigoriev and Faulkner [4]. In that framework, an edge of a finite graph is replaced by a unit interval, and optimization variables may be placed anywhere on the resulting continuous graph. The continuous setting changes the nature of several classical parameters: selected points, covering points, and coloring constraints are no longer restricted to the original vertex set. Consequently, even graph classes that are elementary in the discrete setting may exhibit nontrivial metric packing behavior.

This paper focuses on the distance-two continuous independent-set problem. In the language of ball packing, selected centers must be pairwise at distance at least 2, so the open unit balls centered at selected points do not overlap. Equivalently, following the notation of Grigoriev and Faulkner [4], this is the $r=1$ continuous independent-set parameter ${\alpha }_1$. The complete graph already reveals a new phenomenon: besides choosing original vertices, one may choose midpoints of disjoint edges. This observation suggests a connection with matching theory [2,7].

Our goal is to make this connection exact for complete multipartite graphs. Complete multipartite graphs form a natural test family because their ordinary independence number is trivial, while their continuous geometry contains many edge interiors. We prove that the optimal continuous packing is governed by a simple competition between two support mechanisms: the largest part of the multipartition and a maximum matching. The word “support” is important here: in the matching regime, midpoint configurations give canonical optima, but the theorem does not require or claim uniqueness of midpoint placement.

1.1. Main Contribution and Novelty

Let

$$G=K_{n_1,\dots ,n_k}$$

be a complete multipartite graph with partite sets $V_1,\dots ,V_k$, where $k\ge 2$ and $|V_i|=n_i\ge 1$. Put

$$N={\displaystyle \sum _{i=1}^k}{n}_i,M=\underset{1\le i\le k}{max}{n}_i.$$

The main theorem is the exact formula

$$\begin{array}{|c|}\hline {\alpha }_1(\Gamma \left(K_{n_1,\dots ,n_k}\right))=max\left\{M,⌊N/2⌋\right\}.\\ \hline\end{array}$$

This solves the $r=1$ continuous independent-set problem for the complete multipartite family. The proof is structural. If a feasible set contains a vertex, the metric constraints force all selected points to be vertices from a single part. If a feasible set contains no vertex, then the supporting edges of its points form a matching. These alternatives give the upper bound. The lower bound is achieved by selecting either all vertices in a largest part or, in the matching regime, the midpoints of the edges of a maximum matching. Thus the midpoint construction is used as a canonical certificate of optimality, not as a uniqueness assertion.

The result is useful for three reasons. First, it gives a closed form for a non-discrete metric packing parameter. Second, it shows precisely when the continuous parameter exceeds the ordinary independence number. Third, it yields an exact constructive algorithm that outputs not only the optimum value but also an explicit optimal configuration in the continuous graph.

1.2. Organization

Section 2 introduces the metric-graph model, continuous 1-independence, and the matching number of complete multipartite graphs. Section 3 proves the exact formula. Section 4 presents the constructive algorithm and its complexity. Section 5 gives reproducible numerical illustrations. Section 6 concludes with limitations and possible extensions.

2. Preliminaries

All graphs in this paper are finite, simple, and undirected. If $G=(V,E)$ is a finite graph, its continuous graph $\Gamma \left(G\right)$ is obtained by replacing each edge $uv\in E$ by a closed interval of length one and identifying its endpoints with u and v. The distance between two points $x,y\in \Gamma \left(G\right)$ is the shortest-path distance in this metric graph and is denoted by $\mathrm{dist}(x,y)$.

Definition 1

(Continuous 1-independence). Let Γ be a continuous graph. A subset $S\subseteq \Gamma$ is called 1-independent if

$$\mathrm{dist}(x,y)\ge 2$$

for all distinct $x,y\in S$. The maximum cardinality of such a set is denoted by ${\alpha }_1(\Gamma )$.

This convention corresponds to the radius-one ball-packing interpretation: centers are feasible exactly when their pairwise distances are at least 2. The original vertices of G are also points of $\Gamma \left(G\right)$, so vertex-only independent sets are allowed but are not the only possible feasible sets.

The ordinary independence number of a finite graph G is denoted by $\alpha \left(G\right)$. For a complete multipartite graph $K_{n_1,\dots ,n_k}$,

$$\alpha \left(K_{n_1,\dots ,n_k}\right)=\underset{i}{max}{n}_i=M.$$

We also use the matching number $\nu \left(G\right)$, the maximum number of pairwise vertex-disjoint edges in G.

Lemma 1

(Matching number of a complete multipartite graph). Let $G=K_{n_1,\dots ,n_k}$ with $k\ge 2$, $n_i\ge 1$, $N={\sum }_i{n}_i$, and $M={max}_i{n}_i$. Then

$$\nu \left(G\right)=min\left\{⌊N/2⌋,N-M\right\}.$$

Proof. 

The two upper bounds are immediate. A matching with q edges uses $2q$ vertices, so $q\le ⌊N/2⌋$. If $V_a$ is a largest part, then every edge incident with a vertex of $V_a$ must use a second endpoint outside $V_a$. Since there are only $N-M$ vertices outside $V_a$, no matching can have more than $N-M$ edges. Hence

$$\nu \left(G\right)\le min\left\{⌊N/2⌋,N-M\right\}.$$

We now prove attainability. If $M>N/2$, match every vertex outside a largest part to a distinct vertex inside that largest part. This gives $N-M$ pairwise disjoint edges, so $\nu \left(G\right)=N-M$ in this case.

It remains to consider $M\le N/2$. We claim that G has a perfect matching when N is even and a near-perfect matching when N is odd. Order the current part sizes as

$$a_1\ge a_2\ge \dots \ge a_t>0.$$

While at least two nonempty parts remain, choose vertices from two largest parts and match them. Initially $a_1\le N/2$. Suppose that before one step the remaining total is r and the largest part has size at most $r/2$. After removing one vertex from two largest parts, the new total is $r-2$. The new largest part is at most

$$max\{a_1-1,a_3\},$$

where $a_3=0$ if fewer than three parts are nonempty before the step. Clearly $a_1-1\le (r-2)/2$. If $a_3>(r-2)/2$, then $2a_3>r-2$, hence by integrality $2a_3\ge r-1$. Since $a_1\ge a_2\ge a_3$, we also have $r\ge a_1+a_2+a_3\ge 3a_3$. Combining these two inequalities gives $r\le 3$. Thus the only possible failure of the invariant occurs at the terminal stage, where at most one vertex remains and no further edge is needed. Therefore the greedy procedure continues until either zero or one vertex remains, and it constructs exactly $⌊N/2⌋$ disjoint edges.

The two cases prove the formula.    □

Remark 1.

Lemma 1 is a standard special case of matching theory; see, for example, Lovász and Plummer [7]. The proof is included to make the paper self-contained and to justify the constructive algorithm used below.

3. Exact Formula

We now prove the main theorem. Before the proof, it is useful to keep three small examples in mind:

$${\alpha }_1(\Gamma \left(K_{5,1,1}\right))=5,{\alpha }_1(\Gamma \left(K_{4,2,2}\right))=4,{\alpha }_1(\Gamma \left(K_{3,3,2}\right))=4.$$

The first is governed by the largest part, the second is an equality case in which both canonical constructions have the same size, and the third is governed by a maximum matching.

Theorem 1.

Let $G=K_{n_1,\dots ,n_k}$ be a complete multipartite graph with $k\ge 2$ and $n_i\ge 1$. Put

$$N={\displaystyle \sum _{i=1}^k}{n}_i,M=\underset{1\le i\le k}{max}{n}_i.$$

Then

$${\alpha }_1(\Gamma \left(G\right))=max\left\{M,⌊N/2⌋\right\}.$$

Proof. 

We prove the lower and upper bounds separately.

For the lower bound, let $V_a$ be a largest part. If $u,v\in V_a$ are distinct, then u and v are not adjacent, but since $k\ge 2$ there exists a vertex w outside $V_a$, and $u-w-v$ is a path of length 2. No shorter path exists because u and v are not adjacent. Hence $\mathrm{dist}(u,v)=2$, and $V_a$ is 1-independent in $\Gamma \left(G\right)$. Thus

$${\alpha }_1(\Gamma \left(G\right))\ge M.$$

Next let F be a matching in G, and for each $e\in F$ let $m_e$ denote the midpoint of the corresponding unit interval in $\Gamma \left(G\right)$. If $e=ab$ and $f=cd$ are distinct matching edges, then they have no common endpoint. Any path from $m_e$ to $m_f$ must first travel distance $1/2$ from $m_e$ to one endpoint of e, then traverse at least one full graph edge between an endpoint of e and an endpoint of f, and finally travel distance $1/2$ along f to $m_f$. Therefore

$$\mathrm{dist}(m_e,m_f)\ge {\displaystyle \frac{1}{2}}+1+{\displaystyle \frac{1}{2}}=2.$$

In a complete multipartite graph, such a path of length 2 exists unless the chosen endpoints lie in the same part; in that exceptional endpoint choice, selecting the other endpoint of one matching edge gives a different-part connection because the two edges are disjoint and each joins two different parts. Hence $\mathrm{dist}(m_e,m_f)=2$. The midpoint set $\{m_e:e\in F\}$ is therefore 1-independent, and

$${\alpha }_1(\Gamma \left(G\right))\ge \nu \left(G\right).$$

By Lemma 1, $\nu \left(G\right)=⌊N/2⌋$ whenever $M\le N/2$. If $M>N/2$, then $M>⌊N/2⌋$ and the largest-part construction already gives the larger value. Hence

$${\alpha }_1(\Gamma \left(G\right))\ge max\left\{M,⌊N/2⌋\right\}.$$

For the upper bound, let $S\subseteq \Gamma \left(G\right)$ be any 1-independent set. Suppose first that S contains a vertex $v\in V_a$. Let $x\in S\setminus \left\{v\right\}$. If x is a vertex outside $V_a$, then $\mathrm{dist}(v,x)=1$, impossible. If x is an interior point of an edge $pq$, then at least one endpoint of $pq$ is outside $V_a$ unless both endpoints belonged to $V_a$, which is impossible because no edge lies inside one part. Choose an endpoint $p\notin V_a$. Then $vp$ is an edge, and

$$\mathrm{dist}(v,x)\le 1+\mathrm{dist}(p,x)<2,$$

again impossible. Thus x must be a vertex in $V_a$. Therefore every point of S is a vertex of $V_a$, and $\left|S\right|\le |V_a|\le M$.

Suppose next that S contains no vertex. Then every point of S lies in the interior of an edge of G. No two points of S can lie on the same edge, because their distance would be strictly less than 1. More generally, no two points of S can lie on edges sharing an endpoint. Indeed, if the two supporting edges share a vertex u, then

$$\mathrm{dist}(x,y)\le \mathrm{dist}(x,u)+\mathrm{dist}(u,y)<1+1=2,$$

contradicting 1-independence. Hence the supporting edges of the points of S are pairwise vertex-disjoint; they form a matching in G. Consequently

$$\left|S\right|\le \nu \left(G\right)\le ⌊N/2⌋.$$

The first case gives $\left|S\right|\le M$ and the second gives $\left|S\right|\le ⌊N/2⌋$. Thus every 1-independent set satisfies

$$\left|S\right|\le max\left\{M,⌊N/2⌋\right\}.$$

Combining this with the lower bound proves the theorem.    □

Proposition 1

(Support structure of feasible sets). Let $S\subseteq \Gamma \left(K_{n_1,\dots ,n_k}\right)$ be 1-independent. If S contains a vertex, then S is contained in the vertex set of a single partite class. If S contains no vertex, then the set of graph edges whose interiors contain points of S is a matching.

Proof. 

Both assertions are exactly the two upper-bound cases in the proof of Theorem 1. A selected vertex excludes every edge interior and every vertex outside its own part at distance less than 2. If all selected points are interior points, two supporting edges cannot coincide or share an endpoint, because in either case the two selected points are at distance strictly less than 2 through that common edge or endpoint.    □

Remark 2

(Midpoints are canonical, not unique). The matching construction in Theorem 1 uses edge midpoints because they give an immediate and uniform certificate of feasibility. Proposition 1 is intentionally weaker than a uniqueness theorem: it characterizes the supporting edges but does not claim that every matching-supported optimum must use midpoints. Depending on the multipartite structure and on the chosen matching, other interior placements may also satisfy the distance-two constraint. Conversely, arbitrary non-midpoint placements on a matching need not be feasible. Thus the correct statement is that maximum matchings provide the support of vertex-free optima, while midpoint placement is a canonical optimal realization.

Remark 3

(The edgeless one-part case). If one allows $k=1$, then $K_{n_1}$ has no edges. With the usual convention that distinct connected components have infinite mutual distance, all $n_1$ isolated vertices may be selected, and the formula gives $max\{n_1,⌊n_1/2⌋\}=n_1$. The main proof above assumes $k\ge 2$ only to avoid repeatedly separating this trivial case.

Corollary 1

(Complete graphs). For $n\ge 2$,

$${\alpha }_1(\Gamma \left(K_n\right))=⌊n/2⌋.$$

Proof. 

The complete graph $K_n$ is the complete multipartite graph $K_{1,1,\dots ,1}$, so $N=n$ and $M=1$. Theorem 1 gives the result.    □

Corollary 2

(Complete bipartite graphs). For $a,b\ge 1$,

$${\alpha }_1(\Gamma \left(K_{a,b}\right))=max\{a,b\}.$$

Proof. 

Here $N=a+b$ and $M=max\{a,b\}\ge ⌊(a+b)/2⌋$. Theorem 1 gives the claim.    □

Corollary 3

(Balanced complete multipartite graphs). For $k\ge 2$ and $m\ge 1$,

$${\alpha }_1(\Gamma \left(K_{m,m,\dots ,m}\right))=⌊km/2⌋.$$

In particular,

$${\displaystyle \frac{{\alpha }_1(\Gamma \left(K_{m,m,\dots ,m}\right))}{\alpha \left(K_{m,m,\dots ,m}\right)}}={\displaystyle \frac{⌊km/2⌋}{m}}\sim {\displaystyle \frac{k}{2}}.$$

Proof. 

Here $N=km$ and $M=m$. Since $k\ge 2$, we have $m\le km/2$, and Theorem 1 gives ${\alpha }_1=⌊km/2⌋$. The ratio formula follows because $\alpha \left(K_{m,m,\dots ,m}\right)=m$.    □

4. Algorithmic Construction

Theorem 1 yields an exact constructive algorithm. We call it the Exact Multipartite Continuous Independence Algorithm, abbreviated EMCIA.

For a purely numerical computation, the input is the vector of part sizes $(n_1,\dots ,n_k)$. For a labelled optimal configuration, the input must additionally specify the labelled partite sets

$$V_i=\{v_{i,1},\dots ,v_{i,n_i}\},i=1,\dots ,k.$$

The algorithm returns either vertices $v_{i,j}$ or midpoint labels $m_{uv}$, where $uv$ is an edge of G.

Proposition 2

(Correctness). Algorithm 1 returns an optimal 1-independent set of $\Gamma \left(K_{n_1,\dots ,n_k}\right)$.

Algorithm 1:Exact Multipartite Continuous Independence Algorithm (EMCIA)

Require:  Partite sets $V_i=\{v_{i,1},\dots ,v_{i,n_i}\}$ for $i=1,\dots ,k$. Ensure:  An optimal 1-independent set $S^{★}\subseteq \Gamma \left(K_{n_1,\dots ,n_k}\right)$.  1: $N\leftarrow {\sum }_{i=1}^k{n}_i$.  2: $M\leftarrow {max}_{1\le i\le k}{n}_i$.  3: Let $i^{★}$ be an index with $n_{i^{★}}=M$.  4: if $M\ge ⌊N/2⌋$ then  5:     $S^{★}\leftarrow V_{i^{★}}$.  6: else  7:     Initialize a max-priority queue containing all nonempty parts, keyed by the number of unused vertices.  8:     $F\leftarrow \varnothing$.  9:     while at least two parts in the queue are nonempty do 10:         Extract two parts A and B of largest current size. 11:         Remove one unused vertex u from A and one unused vertex v from B. 12:         $F\leftarrow F\cup \left\{uv\right\}$. 13:         Reinsert A and B if they still contain unused vertices. 14:     end while 15:     $S^{★}\leftarrow \{m_{uv}:uv\in F\}$. 16: end if 17: return $S^{★}$.

Proof. 

If $M\ge ⌊N/2⌋$, Theorem 1 gives ${\alpha }_1(\Gamma \left(G\right))=M$, and all vertices in a largest part form a 1-independent set of size M.

If $M<⌊N/2⌋$, then $M\le N/2$. By Lemma 1, G has a matching of size $⌊N/2⌋$. Moreover, the greedy priority-queue procedure used in Algorithm 1 is exactly the constructive procedure analyzed in the proof of Lemma 1; hence it outputs a maximum matching F of size $⌊N/2⌋$. The midpoint set $\{m_e:e\in F\}$ is 1-independent by the proof of Theorem 1, and its size is optimal. Therefore the algorithm is correct in both regimes.

   □

Remark 4

(Tie handling). When $M=⌊N/2⌋$, both the largest-part construction and a maximum-matching midpoint construction have optimal cardinality. Algorithm 1 returns the largest-part solution in this equality case by convention. This choice is made only to keep the pseudocode deterministic; it is not a uniqueness statement.

4.1. Complexity Analysis

Let $N={\sum }_i{n}_i$. The numerical value

$${\alpha }_1(\Gamma \left(K_{n_1,\dots ,n_k}\right))=max\left\{M,⌊N/2⌋\right\}$$

is computed by a single scan of the part-size vector. Thus the value computation runs in $O\left(k\right)$ time and uses $O\left(1\right)$ auxiliary memory.

For the labelled constructive version, the output can contain $\Theta \left(N\right)$ points, so any algorithm that writes the whole solution requires $\Omega \left(N\right)$ output time. In the largest-part regime, the algorithm outputs $M\le N$ vertices. In the matching-supported regime, the priority queue has at most k elements and the algorithm performs $O\left(N\right)$ extract/reinsert operations, each of cost $O(logk)$. Hence the labelled constructive version runs in

$$O(Nlogk)$$

time and uses $O\left(N\right)$ memory if the output is stored. If the output is streamed, the auxiliary memory beyond the labelled input is $O\left(k\right)$.

Remark 5.

The $O(Nlogk)$ bound is not intended to claim computational difficulty; it is an implementation bound for producing labelled midpoint data. Since the optimum value is available in closed form, the numerical computation itself is linear in the number of parts. If the part sizes are small integers or pre-bucketed, the greedy matching can also be implemented in linear output time.

5. Reproducible Numerical Illustrations

The theorem is exact, so the numerical section is not used as evidence for the formula. Its purpose is to visualize the two extremal regimes and the comparison between the ordinary discrete independence number and the continuous distance-two independence number.

The supplementary code package contains two Python files: emcia.py, which implements the value and constructive algorithms, and generate_experiments.py, which regenerates all CSV files and figures used in this section. The random experiment uses the fixed seed 20260610. It generates 500 complete 5-partite graphs by choosing each part size independently and uniformly from $\{1,\dots ,30\}$.

5.1. Balance Entropy of the Part-Size Vector

A useful quantitative way to describe the transition between the two regimes is to measure how balanced the part-size vector is. Define

$$p_i={\displaystyle \frac{n_i}{N}},H\left(p\right)=-{\displaystyle \sum _{i=1}^k}{p}_{i}log{p}_i,$$

and let

$$h\left(p\right)={\displaystyle \frac{H\left(p\right)}{logk}},N_{\mathrm{eff}}\left(p\right)=exp\left(H\left(p\right)\right).$$

Here $h\left(p\right)\in [0,1]$ is the normalized Shannon entropy and $N_{\mathrm{eff}}$ is the effective number of substantially occupied parts. This entropy is not needed for the proof and is not a complexity-theoretic entropy; it is only a compact numerical diagnostic for the balance of the input multipartition. High entropy indicates that the vertices are spread relatively evenly among the parts, which favors the matching-supported regime. Low entropy indicates concentration in a dominant part, which favors the largest-part regime.

The exact threshold itself remains simpler than the entropy statistic: the largest-part regime occurs precisely when

$${\displaystyle \frac{M}{N}}\ge {\displaystyle \frac{⌊N/2⌋}{N}},$$

whereas the matching-supported regime occurs when $M<⌊N/2⌋$. The entropy statistic is therefore descriptive, not decisive; two vectors can have similar entropy but lie on different sides of the exact threshold.

Table 1. Entropy diagnostics for selected complete multipartite instances. The normalized entropy $h\left(p\right)$ and effective number $N_{\mathrm{eff}}$ quantify balance but do not replace the exact threshold M versus $⌊N/2⌋$.

Graph	$\mathit{M}/\mathit{N}$	$\mathit{h}\left(\mathit{p}\right)$	${\mathit{N}}_{\mathbf{eff}}$	Exact regime
$K_{5,1,1}$	0.714	0.725	2.217	Largest part
$K_{4,2,2}$	0.500	0.946	2.828	Both
$K_{3,3,2}$	0.375	0.985	2.951	Matching-supported
$K_{2,2,2,2}$	0.250	1.000	4.000	Matching-supported
$K_{6,5,5}$	0.375	0.997	2.989	Matching-supported
$K_{10,3,2}$	0.667	0.784	2.365	Largest part

Table 1. Entropy diagnostics for selected complete multipartite instances. The normalized entropy $h\left(p\right)$ and effective number $N_{\mathrm{eff}}$ quantify balance but do not replace the exact threshold M versus $⌊N/2⌋$.

Graph	$\mathit{M}/\mathit{N}$	$\mathit{h}\left(\mathit{p}\right)$	${\mathit{N}}_{\mathbf{eff}}$	Exact regime
$K_{5,1,1}$	0.714	0.725	2.217	Largest part
$K_{4,2,2}$	0.500	0.946	2.828	Both
$K_{3,3,2}$	0.375	0.985	2.951	Matching-supported
$K_{2,2,2,2}$	0.250	1.000	4.000	Matching-supported
$K_{6,5,5}$	0.375	0.997	2.989	Matching-supported
$K_{10,3,2}$	0.667	0.784	2.365	Largest part

5.2. Selected Instances

Table 2 reports representative examples. These examples show the two possible support mechanisms and the transition case where both canonical constructions are optimal.

Figure 1. Comparison of the largest-part value M, the matching value $⌊N/2⌋$, and the exact continuous independence number ${\alpha }_1(\Gamma \left(K_{n_1,\dots ,n_k}\right))$.

Figure 1. Comparison of the largest-part value M, the matching value $⌊N/2⌋$, and the exact continuous independence number ${\alpha }_1(\Gamma \left(K_{n_1,\dots ,n_k}\right))$.

5.3. Random Complete Multipartite Graphs

Figure 2 shows the exact value of ${\alpha }_1$ as a function of the total number of vertices N over the random instances described above. Each point is classified according to whether the optimum is attained by the largest-part construction or by the matching-supported construction.

5.4. Continuous-to-Discrete Comparison Ratio

For a complete multipartite graph, the ordinary independence number is $\alpha \left(G\right)=M$. We therefore define the comparison ratio

$$R\left(G\right)={\displaystyle \frac{{\alpha }_1(\Gamma \left(G\right))}{\alpha \left(G\right)}}={\displaystyle \frac{max\{M,⌊N/2⌋\}}{M}}.$$

This ratio equals 1 in the largest-part-dominant regime and can be larger for balanced multipartite graphs. We use the term comparison ratio rather than relaxation ratio to avoid suggesting that ${\alpha }_1$ comes from a standard mathematical-programming relaxation of $\alpha \left(G\right)$.

For the balanced family $K_{m,m,\dots ,m}$ with k parts, we have $N=km$ and $M=m$. Therefore

$$R\left(K_{m,\dots ,m}\right)={\displaystyle \frac{⌊km/2⌋}{m}}\sim {\displaystyle \frac{k}{2}}.$$

Figure 4 illustrates this linear growth for $m=10$.

Figure 3. Comparison ratio ${\alpha }_1(\Gamma \left(G\right))/\alpha \left(G\right)$ versus the largest part size M over the random instances.

Figure 3. Comparison ratio ${\alpha }_1(\Gamma \left(G\right))/\alpha \left(G\right)$ versus the largest part size M over the random instances.

Figure 4. Comparison ratio for the balanced family $K_{m,\dots ,m}$ with $m=10$.

Figure 4. Comparison ratio for the balanced family $K_{m,\dots ,m}$ with $m=10$.

6. Conclusion

We proved an exact formula for the continuous distance-two independence number of complete multipartite graphs. The formula

$${\alpha }_1(\Gamma \left(K_{n_1,\dots ,n_k}\right))=max\left\{M,⌊N/2⌋\right\}$$

shows that the optimum is governed by two support mechanisms: selecting vertices in a largest part, or using interior points supported on the edges of a maximum matching, with midpoint placement providing a canonical feasible realization. The proof also gives an exact algorithm that computes the value and constructs an optimal labelled configuration.

The method relies essentially on the complete multipartite structure. In a general graph, a vertex-containing continuous independent set need not collapse to one part, and the midpoints of two disjoint edges need not be exactly distance two apart. Thus the reduction to the maximum of a vertex-part term and a matching term is special to this graph family. This limitation also suggests possible directions for further work: identifying graph classes where continuous distance-two independence reduces to matching, deriving formulas for joins or graph blow-ups, and studying whether analogous closed forms exist for continuous vertex cover or continuous coloring parameters.