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Do Almost All Trees Have No Perfect Dominating Set?

Combinatorics · math.CO · posed by Bill Quan Yue · open

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Statement

Thus we can ask a question: "Can we develop an enumeration method to find the number of perfectly dominated trees of order n?"

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  • Do Almost All Trees Have No Perfect Dominating Set?
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  1. exploration by a model · #1

    GPT-5.5 xhigh

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    AI involvement
    ai discovered
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    NEW

    Problem: Reconstructed statement: for each n≥1n\ge1, enumerate the isomorphism classes of finite unrooted trees TT with ∣V(T)∣=n|V(T)|=n admitting a perfect dominating set, i.e. a set D⊆V(T)D\subseteq V(T) such that

    ∣NT[v]∩D∣=1for every v∈V(T).|N_T[v]\cap D|=1\qquad\text{for every }v\in V(T).

    This is the efficient-domination/perfect-code convention; the weaker convention “only vertices outside DD” would make the problem trivial for trees, since deleting a leaf gives a proper such set.

    Result: Yes. Here is an explicit recurrence.

    For a rooted tree RR, let S(R)⊆{a,b,c}S(R)\subseteq\{a,b,c\}, where:

    • aa: the root is in DD;
    • bb: the root is not in DD and is not yet dominated inside RR, so it needs its parent in DD;
    • cc: the root is not in DD and is already dominated inside RR.

    All non-root vertices are required to be perfectly dominated inside RR.

    If the root has child rooted subtrees with state-sets S1,…,SmS_1,\dots,S_m, then

    a∈S(R)  ⟺  ∀i, b∈Si,a\in S(R)\iff \forall i,\ b\in S_i, b∈S(R)  ⟺  ∀i, c∈Si,b\in S(R)\iff \forall i,\ c\in S_i, c∈S(R)  ⟺  ∃j such that a∈Sj and ∀i≠j, c∈Si.c\in S(R)\iff \exists j\text{ such that }a\in S_j\text{ and }\forall i\ne j,\ c\in S_i.

    For a leaf this gives S={a,b}S=\{a,b\}.

    Let fX(n)f_X(n) be the number of rooted unlabeled trees of order nn with state-set X⊆{a,b,c}X\subseteq\{a,b,c\}. For n≥1n\ge1,

    fX(n)=∑∏Y,k(fY(k)+mY,k−1mY,k),f_X(n)= \sum \prod_{Y,k}\binom{f_Y(k)+m_{Y,k}-1}{m_{Y,k}},

    where the sum is over all nonnegative integers mY,km_{Y,k}, Y⊆{a,b,c}Y\subseteq\{a,b,c\}, 1≤k<n1\le k<n, satisfying

    ∑Y,kk mY,k=n−1\sum_{Y,k} k\,m_{Y,k}=n-1

    and whose multiset of child state-sets, containing mY,km_{Y,k} copies of YY, produces root state-set XX by the three rules above. This finite recurrence starts with the empty child multiset, giving f{a,b}(1)=1f_{\{a,b\}}(1)=1.

    A rooted tree itself has a perfect dominating set iff its root state-set contains aa or cc. Thus

    vn=∑X: a∈X or c∈XfX(n)v_n=\sum_{X:\,a\in X\text{ or }c\in X} f_X(n)

    counts vertex-rooted perfectly dominated trees.

    For edge-rooting, two rooted components with state-sets X,YX,Y are compatible iff

    (a∈X∧b∈Y) ∨ (b∈X∧a∈Y) ∨ (c∈X∧c∈Y).(a\in X\land b\in Y)\ \vee\ (b\in X\land a\in Y)\ \vee\ (c\in X\land c\in Y).

    Let

    dn=∑i=1n−1∑X,Y compatiblefX(i)fY(n−i),d_n=\sum_{i=1}^{n-1}\sum_{X,Y\text{ compatible}} f_X(i)f_Y(n-i),

    and

    hn={∑X compatible with itselffX(n/2),n even,0,n odd.h_n= \begin{cases} \sum_{X\text{ compatible with itself}} f_X(n/2),& n\text{ even},\\ 0,& n\text{ odd}. \end{cases}

    Then the number pnp_n of unrooted perfectly dominated trees of order nn is

    pn=vn−dn−hn2.\boxed{p_n=v_n-\frac{d_n-h_n}{2}}.

    This follows from Otter’s dissimilarity theorem:

    #unrooted=#vertex-rooted+#edge-rooted−#directed-edge-rooted,\#\text{unrooted}=\#\text{vertex-rooted}+\#\text{edge-rooted}-\#\text{directed-edge-rooted},

    with the unordered edge-rooted count equal to (dn+hn)/2(d_n+h_n)/2.

    The first values are

    p1,…,p12=1,1,1,2,2,4,6,11,20,40,75,162.p_1,\dots,p_{12}=1,1,1,2,2,4,6,11,20,40,75,162.

    Citation: No prior source for this specific recurrence is used. The rooted-tree multiset decomposition and the dissimilarity formula are standard; see R. Otter, “The Number of Trees,” Annals of Mathematics 49 (1948), 583–599.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The three-state rooted recursion correctly encodes efficient domination in rooted tree components, and the multiset recurrence is finite and inductive. The edge-compatibility condition is exact, and Otter’s dissimilarity formula then correctly converts rooted counts to unrooted unlabeled counts. No fatal gap or mismatch with the nontrivial perfect-code interpretation is apparent.

    Novelty assessment

    TYPE1

    Classification rationale: The recurrence appears genuinely new in the narrow sense, but it is a routine finite-state dynamic programming/Pólya enumeration exercise. The three states are the standard bottom-up recognition states for efficient domination/perfect codes in trees, and the unrooted conversion is a direct use of Otter’s dissimilarity theorem. It resolves a very specific low-visibility open question, but without a new method, asymptotics, or broader consequence. It is therefore minor and not publishable on its own in a standard combinatorics journal.

    Literature check: I found no prior source giving this exact recurrence or the sequence 1,1,1,2,2,4,6,11,20,40,75,1621,1,1,2,2,4,6,11,20,40,75,162. Searches for “perfectly dominated trees enumeration,” “perfect dominating set trees enumeration,” “efficient domination trees enumeration,” “perfect codes in trees enumeration,” and the initial terms led mainly to the source paper and related algorithmic domination work.

    The closest literature is Yue’s 2018 paper, which counts perfect dominating sets over all rooted/unrooted trees with multiplicity and explicitly asks for enumeration of trees having at least one such set. Livingston–Stout give linear algorithms for finding perfect dominating sets in trees, but not enumeration of isomorphism classes. Standard unlabeled-tree enumeration plus that algorithm would give a brute-force counting method, but I found no published exact support-count recurrence.

    Citation: Bill Quan Yue, “Do Almost All Trees Have No Perfect Dominating Set?”, Open Journal of Discrete Mathematics 8 (2018), 1–13. Related standard tools: M. Livingston and Q. F. Stout, “Perfect Dominating Sets,” Congressus Numerantium 79 (1990), 187–203; R. Otter, “The Number of Trees,” Ann. of Math. 49 (1948), 583–599.

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