Tight bounds for generalized power domination in regular graphs

Abstract: Dorbec et al. [SIAM J. Discrete Math., 27 (2013)] conjectured that, for all integers $k\geq1$ and $r\geq3$, every connected $r$-regular graph $G$ of order $n$, other than $K_{r,r}$, satisfies $γ<em>{P,k}(G)\leq n/(r+1)$. After disproving this conjecture, Chen et al.[Graphs Combin., 38 (2022)] proposed a corresponding conjecture for claw-free regular graphs. In this paper, we prove this conjecture: for integers $k\geq\ell\geq1$, every connected claw-free $(k+\ell+1)$-regular graph $G$ of order $n$ satisfies $γ</em>{P,k}(G)\leq n/(k+\ell+2)$, and this bound is tight. Moreover, without the claw-free assumption, we show that, for each fixed integer $k\geq1$, the supremum of $γ_{P,k}(G)/\lvert V(G)\rvert$ over all connected $r$-regular graphs $G$ is asymptotic to $(\ln r)/r$ as $r\to\infty$.

• The paper proves that every connected claw-free (k+ℓ+1)-regular graph with k≥ℓ has a k-power dominating set of size at most n/(k+ℓ+2), and that this bound is tight.
• Its strong-edge decomposition shows that weak neighborhoods are small cliques, allowing one representative per strong component to activate the entire graph through propagation.
• Without claw-freeness, the worst-case ratio satisfies c_k(r)∼(ln r)/r for fixed k, showing that generalized power domination can be asymptotically no easier than ordinary domination.

Background and problem

Power domination originated in the placement of phasor measurement units in electrical networks: Haynes et al. recast the monitoring problem as a graph parameter in which an initial set dominates, and thereafter any monitored vertex with exactly one unmonitored neighbor monitors that neighbor [(1502.00000)-style citation to HaynesEtAl2002]. Chang et al. generalized this to $k$-power domination, allowing a monitored vertex with at most $k$ unmonitored neighbors to monitor all of them; $k=0$ recovers domination and $k=1$ classical power domination. For a connected $(k+1)$-regular graph one vertex suffices, so the natural regime is degree $r=k+\ell+1$ with $\ell\ge 2$.

Dorbec et al. proved that every connected $(k+2)$-regular graph other than $K_{k+2,k+2}$ satisfies $\gamma_{P,k}(G)\le n/(k+3)$, and conjectured that for all $k$0, $k$1, every connected $k$2-regular graph $k$3 of order $k$4 satisfies $k$5. This conjecture was subsequently disproved across its entire nontrivial range: Lu et al. gave counterexamples for $k$6 and every even $k$7, and Chen et al. constructed counterexamples for every $k$8 and $k$9. Chen et al. further showed that even within claw-free $k=0$0-regular graphs the bound can fail when $k=0$1, which motivated their restricted conjecture: writing $k=0$2, for all integers $k=0$3, every connected claw-free $k=0$4-regular graph of order $k=0$5 satisfies $k=0$6, with tight bound. The cases $k=0$7 were previously known; the paper under review settles the conjecture completely.

Main theorem: sharp claw-free bound

The main result confirms the claw-free conjecture in full.

Theorem (main). For integers $k=0$8, every connected claw-free $k=0$9-regular graph $k=1$0 of order $k=1$1 satisfies $k=1$2, and the bound is tight.

The proof rests on a decomposition by strong edges: an edge $k=1$3 is strong if $k=1$4, weak otherwise. The key structural lemma shows that in a claw-free $k=1$5-regular graph with $k=1$6 and $k=1$7, the weak neighborhood $k=1$8 of any vertex is a clique of size at most $k=1$9, so each vertex has at least $(k+1)$0 strong neighbors. The argument is a counting contradiction in the complement of $(k+1)$1: that complement must be triangle-free (else a claw arises), which forces $(k+1)$2 for nonadjacent $(k+1)$3, while the definition of weak edges forces $(k+1)$4 since $(k+1)$5.

Two consequences drive the proof. First, for any strong component $(k+1)$6 and $(k+1)$7, the outside-neighborhood $(k+1)$8 is a clique. Second, a single vertex activates its whole strong component under propagation: along a strong edge $(k+1)$9, once $r=k+\ell+1$0 is monitored, $r=k+\ell+1$1 has at least $r=k+\ell+1$2 monitored neighbors, hence at most $r=k+\ell+1$3 unmonitored ones, so propagation continues through the strong subgraph.

The covering argument then proceeds as follows. Take an inclusion-minimal cover $r=k+\ell+1$4 of strong components such that every vertex lies in the closed neighborhood of some member. Minimality supplies, for each $r=k+\ell+1$5, a representative $r=k+\ell+1$6 whose closed neighborhood meets no other member of $r=k+\ell+1$7; the clique property of outside-neighborhoods then implies these representatives have pairwise disjoint closed neighborhoods, giving $r=k+\ell+1$8. Selecting one representative per component yields a $r=k+\ell+1$9-PD-set of size $\ell\ge 2$0. Tightness follows from constructions of Chen et al., so the constant $\ell\ge 2$1 cannot be improved. This resolves the threshold phenomenon identified earlier: the conjectured bound holds for claw-free regular graphs exactly when $\ell\ge 2$2, i.e., $\ell\ge 2$3.

Asymptotics without the claw-free assumption

Without claw-freeness, the exact best upper bound on $\ell\ge 2$4 in connected $\ell\ge 2$5-regular graphs remains open for $\ell\ge 2$6 — a problem Dorbec posed explicitly. The paper addresses it asymptotically via the ratio

$\ell\ge 2$7

and proves that for each fixed $\ell\ge 2$8, $\ell\ge 2$9 as $(k+2)$0. More precisely, with $(k+2)$1 and $(k+2)$2, for every $(k+2)$3 and large $(k+2)$4,

$(k+2)$5

Upper bound. Every dominating set is a $(k+2)$6-PD-set, so the Caro–Roditty bound $(k+2)$7 applied at minimum degree $(k+2)$8 gives the upper bound directly. Since $(k+2)$9, this already matches the asymptotic order.

Lower bound. The construction is a lexicographic-type blow-up. Starting from a connected $K_{k+2,k+2}$0-regular graph $K_{k+2,k+2}$1 with $K_{k+2,k+2}$2 (Alon–Wormald), replace each vertex by a copy of an $K_{k+2,k+2}$3-regular graph $K_{k+2,k+2}$4 on $K_{k+2,k+2}$5 vertices ($K_{k+2,k+2}$6), joining copies completely according to edges of $K_{k+2,k+2}$7. The result is connected and $K_{k+2,k+2}$8-regular. Each copy $K_{k+2,k+2}$9 is a $\gamma_{P,k}(G)\le n/(k+3)$0-fort: any external neighbor of $\gamma_{P,k}(G)\le n/(k+3)$1 is adjacent to all $\gamma_{P,k}(G)\le n/(k+3)$2 of its vertices, so by the fort obstruction (any $\gamma_{P,k}(G)\le n/(k+3)$3-PD-set must meet $\gamma_{P,k}(G)\le n/(k+3)$4 for every $\gamma_{P,k}(G)\le n/(k+3)$5-fort $\gamma_{P,k}(G)\le n/(k+3)$6), a $\gamma_{P,k}(G)\le n/(k+3)$7-PD-set must hit $\gamma_{P,k}(G)\le n/(k+3)$8 for every $\gamma_{P,k}(G)\le n/(k+3)$9. The set of indices whose copies are hit dominates $k$00, whence $k$01, yielding the lower bound.

A notable consequence: had the original Dorbec et al. conjecture been true, one would have $k$02 for large $k$03 (attained by $k$04). Instead, unrestricted $k$05-power domination in regular graphs is asymptotically as hard as domination, with ratio growing like $k$06 rather than $k$07 — a polynomial-factor separation driven entirely by graphs containing claws.

Limitations and open questions

The sharp bound requires both structural hypotheses simultaneously: claw-freeness and the balance condition $k$08 (equivalently $k$09). Both are necessary — counterexamples exist without either — but the paper leaves open the precise value or best upper bound of $k$10 for fixed $k$11 and finite $k$12 in the range $k$13, including whether the lower-bound construction's constant factor can be matched. The asymptotic result fixes only the leading term $k$14; the second-order behavior of $k$15, and the dependence of the implicit constants on $k$16, remain undetermined.

Conclusion

The paper establishes two complementary results on generalized power domination in regular graphs. First, it fully proves the Chen–Lu–Ye conjecture: connected claw-free $k$17-regular graphs admit $k$18-PD-sets of size at most $k$19, tightly, via a clean strong-edge decomposition argument. Second, it shows that once the claw-free restriction is dropped, the worst-case $k$20-power domination ratio over connected $k$21-regular graphs grows as $k$22 for every fixed $k$23, matching the domination number's asymptotics and quantifying how much the failed general conjecture was off. Together these results delineate precisely where propagation-based monitoring retains a linear advantage over ordinary domination and where that advantage vanishes.