Paths with Prescribed Endpoints in Semicomplete and Locally Semicomplete Digraphs

Abstract: We study two open path problems with prescribed endpoints posed by Bang-Jensen and Gutin. The first asks for a longest $(x,y)$-path in a semicomplete digraph. The second asks whether a locally semicomplete digraph has a Hamiltonian $(x,y)$-path. For semicomplete digraphs, we solve the first problem when the endpoints lie in different strong components. We also prove that if a non-Hamiltonian longest $(x,y)$-path omits a set of vertices, then these vertices together with $x$ and $y$ have a Hamiltonian $(y,x)$-path. This gives an equivalent cycle problem. We then give an exact algorithm that runs in polynomial time when the number of omitted vertices is fixed. For locally semicomplete digraphs, we determine the possible endpoints in the connected nonstrong case. Known results then leave only strong, nonsemicomplete, non-$4$-strong digraphs unresolved. Every such digraph of order at least five has a strong vertex cut of size at most three. Two examples show that a spanning directed path together with a vertex-disjoint directed cycle is not sufficient, and that vertices of one strong component need not occur consecutively on a Hamiltonian path.

• The paper proves that every non-Hamiltonian longest (x,y)-path in a strong semicomplete digraph lies on a Hamiltonian cycle, reducing optimization to finding a maximum cycle through a prescribed arc.
• The paper develops an exact algorithm running in XP time when parameterized by the number of omitted vertices, while leaving fixed-parameter tractability and the general polynomial-time case open.
• The paper characterizes Hamiltonian (x,y)-paths in connected nonstrong locally semicomplete digraphs and isolates the remaining strong, nonsemicomplete, non-4-strong case, supported by counterexamples that challenge small-cut dynamic programming.

Overview

This paper by Bai, Yan, and Zhou addresses two open problems posed by Bang-Jensen and Gutin concerning directed paths with prescribed endpoints. The first asks whether a longest $(x,y)$-path in a semicomplete digraph can be found in polynomial time; the second asks whether Hamiltonian $(x,y)$-paths in locally semicomplete or extended semicomplete digraphs can be decided in polynomial time. The paper makes partial progress on both, delivering structural characterizations, an $\mathsf{XP}$-type exact algorithm, a complete solution of the connected nonstrong locally semicomplete case, and two counterexamples that delimit what any general algorithm can exploit.

Longest paths in semicomplete digraphs

The paper first settles the case where the prescribed endpoints lie in different strong components. The condensation of a semicomplete digraph is an acyclic tournament, hence has a unique topological ordering $C_1,\ldots,C_t$ with all arcs between components directed forward. If $x\in C_i$ and $y\in C_j$ with $i<j$, then every longest $(x,y)$-path has vertex set exactly $V(C_i)\cup\cdots\cup V(C_j)$, and such a path is constructible in polynomial time by concatenating Hamiltonian paths of the individual components. If $i>j$ no $(x,y)$0-path exists, and if $(x,y)$1 every $(x,y)$2-path lies inside $(x,y)$3.

The central structural contribution concerns strong semicomplete digraphs. The paper proves that if $(x,y)$4 is a non-Hamiltonian longest $(x,y)$5-path in a strong semicomplete digraph, then the induced subdigraph on the omitted vertices $(x,y)$6 together with $(x,y)$7 and $(x,y)$8 admits a Hamiltonian $(x,y)$9-path. The proof partitions $\mathsf{XP}$0 into $\mathsf{XP}$1 and $\mathsf{XP}$2 (shown to cover $\mathsf{XP}$3 by an insertion argument against maximality of $\mathsf{XP}$4), then assembles the $\mathsf{XP}$5-path from Hamiltonian paths of the strong components of $\mathsf{XP}$6, using a Hamiltonian cycle of $\mathsf{XP}$7 in the case of a single component. The immediate corollary is that every non-Hamiltonian longest $\mathsf{XP}$8-path in a strong semicomplete digraph is a directed subpath of a Hamiltonian cycle. Via a straightforward arc-addition reduction, the longest $\mathsf{XP}$9-path problem is therefore equivalent to finding a maximum-order directed cycle containing one prescribed arc.

Algorithmic consequences and their limits

Combining the structural result with the Bang–Jensen–Manoussakis–Thomassen polynomial-time Hamiltonian $C_1,\ldots,C_t$0-path algorithm for semicomplete digraphs, the paper obtains an exact algorithm that enumerates candidate omitted sets of size up to $C_1,\ldots,C_t$1 and runs in time

$C_1,\ldots,C_t$2

which is polynomial for every fixed $C_1,\ldots,C_t$3 and places the problem in $C_1,\ldots,C_t$4 when parameterized by $C_1,\ldots,C_t$5. The authors are explicit that this is not an $C_1,\ldots,C_t$6 bound: the exponent of the polynomial depends on $C_1,\ldots,C_t$7, and the paper does not claim fixed-parameter tractability. A remark also cautions that the polynomial directed $C_1,\ldots,C_t$8-linkage algorithm for semicomplete digraphs does not yield the required optimization (two prescribed paths covering $C_1,\ldots,C_t$9 with one of minimum order), so no polynomial-time algorithm for the full conjecture follows from existing machinery.

Locally semicomplete digraphs

For connected nonstrong locally semicomplete digraphs, the paper gives a complete answer. Using Bang-Jensen's decomposition theorem — the strong components admit a topological ordering $x\in C_i$0 with $x\in C_i$1 for all $x\in C_i$2 — it proves that a Hamiltonian $x\in C_i$3-path exists if and only if $x\in C_i$4 and $x\in C_i$5, and that such a path can be constructed in polynomial time when these conditions hold. Necessity follows because component indices along a directed path are nondecreasing.

Combining this with the disconnected case (trivially no), the semicomplete case (the 1992 algorithm), and Guo's theorem that every $x\in C_i$6-strong locally semicomplete digraph is strongly Hamiltonian-connected, the paper shows that any unresolved instance of the second conjecture may be assumed strong, nonsemicomplete, and non-$x\in C_i$7-strong. Every such digraph of order at least five has a strong vertex cut of size at most three, findable in polynomial time by exhaustive enumeration of small vertex sets. The authors note a caveat: Guo's theorem is quoted as an existence statement, and the paper does not derive a polynomial-time construction algorithm from it.

Counterexamples

Two small constructions delimit the approach via small strong vertex cuts. The first is a four-vertex strong semicomplete digraph that contains a spanning subdigraph consisting of an $x\in C_i$8-path plus a vertex-disjoint directed cycle, yet has no Hamiltonian $x\in C_i$9-path — because the prescribed start vertex has outdegree one. This shows that the spanning path-plus-cycle certificate is insufficient in general; the digraph is not $y\in C_j$0-strong, so it does not refute the extended-tournament conjecture of Bang-Jensen and Gutin.

The second is a five-vertex strong, nonsemicomplete, non-$y\in C_j$1-strong locally semicomplete digraph with a unique Hamiltonian $y\in C_j$2-path, in which the two vertices of one strong component of $y\in C_j$3 (for $y\in C_j$4 a single vertex) are separated on the path by vertices of other components. Consequently, any dynamic program over a small strong vertex cut must track multiple entries into and exits from each strong component, and the known decomposition theorem gives no polynomial bound on the number of such ordered lists. This is the paper's explanation of why the size-at-most-three cut does not immediately yield a polynomial-time algorithm.

Limitations and open problems

The paper is explicit about what remains unresolved. Conjecture 1 (polynomial-time computation of longest $y\in C_j$5-paths in semicomplete digraphs) is settled only when $y\in C_j$6 and $y\in C_j$7 lie in different strong components or when the number of omitted vertices $y\in C_j$8 is bounded. The $y\in C_j$9 algorithm leaves open whether the problem is fixed-parameter tractable parameterized by $i<j$0. The extended-tournament conjecture on $i<j$1-strong digraphs with a cycle factor in the remainder is not addressed. For locally semicomplete digraphs, the strong, nonsemicomplete, non-$i<j$2-strong case — precisely the case admitting a strong vertex cut of size at most three — remains open, and the counterexample above indicates that the cut alone does not determine the structure of Hamiltonian paths.

Conclusion

The paper converts two endpoint-prescribed path conjectures into more tractable forms: the longest-path problem becomes a maximum cycle problem through a prescribed arc, with a polynomial solution when few vertices are omitted; and the Hamiltonian-path decision problem for locally semicomplete digraphs is reduced to a single strong case characterized by a small strong vertex cut. The two counterexamples sharpen the remaining difficulty, showing both that spanning path-plus-cycle decompositions cannot be merged under prescribed endpoints and that component intervals on Hamiltonian paths need not be contiguous. The open problems — $i<j$3 status of the omitted-vertices parameter and polynomial decidability in the remaining strong case — are now stated in narrower, structurally explicit terms.