In this paper we determine the metric dimension of K_a × K_b × K_c for all a,b,c∈ℕ with a ≤ b ≤ c as follows. For 3a c , it is ⌊2/3(b+c-1) ⌋ , and for 3a=b+c , it is ⌊a+b+c/2⌋ -1 . The only open case is 3a>b+c , where two values are possible, namely ⌊a+b+c/2⌋ -1 and ⌊a+b+c/2⌋ . This result extends previous results of [4], who computed the metric dimension of K_a × K_b , and of [14], who computed the metric dimension of K_a × K_a ×K_a . We prove our result by introducing and analyzing a new variant of Static Black-Peg Mastermind, in which each peg has its own permitted set of colors. For all cases, we present strategies which we prove to be both feasible and optimal. Our main result follows, as the number of questions of these strategies is equal to the metric dimension of K_a × K_b × K_c .
The k-Cycle Perfect Matching Reconfiguration Problem (k-Cycle-PM-RP) concerns the task of deciding if a pair of perfect matchings for a simple graph can be interconverted via a series of local rearrangements that allow for transitions between perfect matchings if and only if the symmetric difference of their edge sets induces a simple cycle of length k ∈ 2ℕ_>1 (Bonamy et al.; Proc. 44th MFCS; 2019). While it is known that the k-Cycle-PM-RP is PSPACE-complete in the case of bounded bandwidth and bounded degree bipartite graphs, as well as polynomial time solvable in the case of outerplanar graphs, cographs, and strongly orderable graphs, it has remained an open question whether the problem is fixed-parameter tractable in the genus of a graph. In this work, we answer this question by showing that the k-Cycle-PM-RP remains PSPACE-complete for bounded bandwidth and bounded degree planar graphs for every k ∈ 2ℕ_>1 . On the other hand, among some other positive results, we show that it is fixed-parameter tractable in a parameter r ∈ℕ to add at most r edges to a planar graph G to construct a strongly orderable planar graph G' , where we can observe that the ( k=4) -Cycle-PM-RP on strongly orderable graphs is both polynomial time tractable and guaranteed to admit a positive solution.
For a graph G, a set S⊆ V(G) is a cosecure dominating set if S is a dominating set of G and for each v ∈ S there exists a vertex v' ∈ N(v) ∖ S such that (S ∪{v'}) ∖{v} is also a dominating set. Given a graph G, Cosecure Domination (Co-SDS, in short) asks to find a minimum cosecure dominating set for G. The problem is W[2]-hard parameterized by the solution size. Kusum and Pandey (Theor. Comput. Sci., 2024) showed that Co-SDS is linear time solvable on graphs of bounded treewidth and cliquewdith by using monadic second order logic. In this paper, we present explicit algorithms for the problem with respect to structural parameters like modular width, treewidth, distance to cluster, distance to co-cluster, and vertex cover number. Here, we want to emphasize that the running times of all our FPT algorithms are single exponential or only slightly super-exponential. We further show that Co-SDS when parameterized by vertex cover number, does not admit a polynomial kernel unless 𝖼𝗈𝖭𝖯 ⊆ 𝖭𝖯/𝗉𝗈𝗅𝗒 .
We introduce the Red-Blue Unshared Domination (RBUD) problem, a variant of domination problems in graphs. Given an undirected graph G with vertices colored either red or blue, the goal is to find a minimum-size subset of vertices S such that for every red-blue pair of vertices outside S, there exist two distinct vertices in S (called unshared dominators), each dominating one vertex of the pair but not the other. Specifically, for each red vertex r ∉S and blue vertex b∉S , there must be a vertex in S that dominates r but not b and another vertex in S that dominates b but not r. We prove that the RBUD problem is NP-hard, even for bipartite planar subcubic graphs with large girth. We also provide an approximation algorithm for general graphs and exact efficient algorithms for trees and graphs with bounded neighborhood diversity. Additionally, we extend our algorithmic approach to graphs with bounded modular-width.
Rotation distance is a fundamental parameter in the study of tree balancing and data structures. One of the fundamental open problem in the area is to compute the minimum number of rotations required to transform one tree to the other, in which both trees have same number of nodes. We define shallow rotation distance with parameter k (which we call k-shallow rotation distance, denoted by d_k(T_1,T_2) ) to be the rotation distance between any two full binary trees when the rotation is allowed only at nodes within depth k. Indeed, 2-shallow rotation distance is well defined and is already the frontier of unknown territory in algorithmic approaches to rotation distance problem. In this paper, we identify a special case of rotation distance problem, known as the left- λ - RotDist and right- λ - RotDist and prove that they are as hard as the general version of the problem. Complementing this, we give a polynomial time algorithms for computing the d_2(T_1,T_2) in three scenarios (a) when T_1 is a right- λ -tree and T_2 is a right-comb tree, (b) when T_1 is a left- λ tree and T_2 is a left-comb-tree and (c) when T_1 is a left-right- λ -tree and T_2 is a double comb with matching subtree size, where a double-comb is a tree in which all the nodes are present only on the right-most or left-most path from the root. We obtain our results by using a known connection between the rotation distance and the well-studied algebraic object called Thompson’s group ℱ [8]. We use the forest representation for a pair of binary trees introduced by Belk and Brown (2005), where we design a new measure for pairs of spaces in the representation, and demonstrate that this new measure indeed captures the shortest representation of the group element in the above three scenarios.
In the Connected Cluster Vertex Deletion problem, the input is a graph G and an integer k , and the task is to determine whether there exists a vertex set S ⊆ V(G) of size at most k such that G[S] is connected and the graph G - S is a cluster graph—that is, every connected component of G - S is a clique. The problem is known to be NP-complete and has been previously studied from the viewpoint of classical computational complexity. In this paper, we initiate the study of Connected Cluster Vertex Deletion from the perspective of parameterized complexity. We show that the problem is fixed-parameter tractable (FPT) when parameterized by the solution size k , by designing an algorithm with running time 𝒪(5.2^k · n^𝒪(1) ) . Furthermore, we show that Connected Cluster Vertex Deletion does not admit a polynomial kernel unless ⊆/ . To cope with this kernelization lower bound, we complement our result by designing a polynomial-size α -approximate kernel for any fixed α > 1 .
Dominance is a fundamental concept in graph theory. Motivated by applications in facility location problems, we study colorful k-rainbow domination. Given a graph G = (V, E) and a set of k colors, we consider a function f that assigns a subset of colors to each vertex. This function is called the colorful k-rainbow dominating function of G if two conditions hold. First, for each vertex v, its closed neighborhood N [v] contains all k different colors. Second, each vertex v gets a non-empty color set. The weight of f is the total number of colors assigned. The goal of the colorful k-rainbow domination problem is to find a colorful k-rainbow dominating function with the smallest weight, called the colorful k-rainbow domination number. Colorful 3-rainbow domination is NP-complete in general graphs, but solvable in linear time for trees. We give linear time algorithms that compute an optimal colorful 3-rainbow domination function if an input graph is a block or a cactus graph. Furthermore, we show that the corresponding problem remains NP-complete if we restrict the input graph to the class of maximum degree 3 graphs.
A quantum analogue of the classical first-order logic (FO) was discussed lately in [CiE 2024, LNCS 14773, pp. 311–323] from a semantic viewpoint in order to characterize logarithmic-time/space quantum computability. From a syntactic perspective, we further study this new, intriguing quantum logic by introducing a quantum analogue of Gentzen’s tree-like proof system of classical natural deduction and further analyzing the expressing power of this new quantum natural deduction system. In particular, we prove the fundamental features of the soundness and completeness of this quantum proof system and, as a consequence of them, we further show the compactness of the system. Quantum natural deduction is expected to pave a road to automated theorem proving in a quantum-mechanical fashion.
The Positive Influence Domination (PID) problem asks to find a minimum cardinality subset of influencers among the vertices of an undirected graph such that at least half the number of neighbors of each vertex are influencers. The problem's underlying model can be used to determine an economical way of promoting (and keeping) good habits in society by interpreting vertices as individuals, neighbors as social contacts, and influencers as, for example, healthy eaters. We show that the problem is NP-hard even when restricted to planar subcubic graphs. The same result turns out to apply for the so-called double total domination problem, which exhibits a similar behavior on this graph class. We use this insight to derive NP-hardness of PID on the class of induced partial grids via a technique relying on orthogonal graph drawing. Finally, we derive bounds on the size of optimal solutions for arbitrarily dimensioned grids.
For a Boolean circuit C, the energy complexity of C is defined as the number of gates outputting ones in C, where the maximum is taken over all the input assignments. The energy complexity has been well-studied among various bases since 1960s. Monotone circuits incur high energy in terms of their size. In order to understand how much energy can be saved by using negations in the circuit design, in this paper, we study the contrast between two infinite bases ℬ_* and 𝒩_* , where ℬ_* contains the conjunction of unbounded fan-in and disjunction of unbounded fan-in, and the negation, while 𝒩_* contains the conjunction and disjunction with unbounded fan-in, where any input can be negated. Thus, a ℬ_* -circuit needs a single gate for negation, while 𝒩_* -circuit can freely use negation without contributing to the energy of the circuit. We show that this “free” use of negations has a considerable impact on computational power of energy-bounded circuits, as follows: The results above show equipping gates with negations are helpful for energy-bounded circuits. On the limitation of 𝒩_* -circuits, we show:
This paper studies the disjointness, the inclusion, and the regularity problem for nonterminating behaviors of concurrent systems modelled using Mazurkiewicz traces. The main finding is a trichotomy that, for every independence alphabet, determines the complexity of the problems (almost) completely. Noteworthy, for all the problems, the classes are the same. While these classes already appeared in the study of terminating behaviors, the trichotomy obtained here is more pronounced as the complexities vary from decidable via low levels of the arithmetical hierarchy to low levels of the analytical hierarchy (that do not feature in the result for terminating behaviors).
This paper investigates the computational complexity of Locally Rainbow Path on undirected graphs. In this problem, given a vertex-colored graph with q colors, two vertices s, t, and two integers r,ℓ , we seek an s-t path of length at most ℓ where every subpath of length at most r contains vertices with pairwise distinct colors. Here, the length of a path is defined by the number of vertices in the path. We establish the complexity boundaries by proving that Locally Rainbow Path is NP-complete when r ≥ 4 and q ≥ 5 , while tractable for r ≤ 2 or when r = 3 and q ≤ 3 . For the case where r = q , we provide an r^O(r)(n+m) -time algorithm, making it fixed-parameter tractable when parameterized by r. From a structural perspective, we prove the paraNP-completeness for pathwidth on cactus graphs of maximum degree 3, while showing fixed-parameter tractability when parameterized by treewidth plus the number of colors or by vertex cover number.
The Strongly Connected Steiner Subgraph (SCSS) problem only demands strong connectivity between a set of terminals: nothing is specified about how many (disjoint) paths are required between each pair of terminals. Chitnis et al. [IPEC ’14, Algorithmica ’17] introduced a variant of SCSS (on two terminals) with demands for the number of paths between the two terminal pairs. Formally, the 2 -SCSS- (k_1,k_2) problem is defined as follows: given an edge-weighted directed graph G=(V,E) on n vertices with weight function ω : E→ℝ^≥ 0 , two terminal vertices s, t, and integers k_1, k_2, p ; the question is whether there exists a set of k_1 paths F_1, F_2, … , F_k_1 from s t and k_2 paths B_1, B_2, … , B_k_2 from t s such that ∑ _e∈ Eω (e)·ϕ (e)≤ p , where ϕ (e)= max{|{i∈ [k_1] : e∈ F_i}| , |{j∈ [k_2] : e∈ B_j}|} . For each k≥ 1 , Chitnis et al. [IPEC ’14, Algorithmica ’17] designed a n^O(k) time algorithm for graphs with n vertices, and obtained a matching lower bound of f(k)· n^o(k) under the Exponential Time Hypothesis (ETH) for any computable function f. In this paper, we address the next natural questions for the 2 -SCSS- (k,1) problem: As a consequence of our results, we also obtain W[1]-hardness of the 2 -SCSS- (k,1) problem on general graphs parameterized by the cost p of the solution: this does not follow from the W[1]-hardness proof of Chitnis et al. [IPEC ’14, Algorithmica ’17] since the value of p in their reduction was a function of both k and n.
We exhibit an ^3 isomorphism test for Fitting-free groups given by their Cayley tables, a class for which isomorphism testing was previously known to be in (Babai, Codenotti, Qiao; ICALP ’12). In sharp contrast, we show that for permutation groups, isomorphism testing of Fitting-free groups is at least as hard as Graph Isomorphism and Linear Code Equivalence (the latter being GI-hard and having no known subexponential-time algorithm). Lastly, we show that any Fitting-free group of order n is identified by formulas (without counting) using only O(loglog n) variables. This is in contrast to the fact that there are infinite families of Abelian groups that are not identified by FO formulas with o(log n) variables (Grochow Levet, FCT ’23).
Horiyama, Kobayashi, Ono, Seto, and Suzuki introduced the pre-assignment model for uniquifying the optimal solutions of a given instance of optimization problems [AAAI 2024]. For example, a pre-assignment of minimum vertex cover C is defined as a vertex set C_pre=(C_in, C_ex) such that C includes vertices in C_in and excludes ones from C_ex . The pre-assignment problem for unique minimum vertex covers (PAU-VC) asks whether there exists a pre-assignment of at most k vertices that induces a unique minimum vertex cover of a given graph. PAU-VC has three pre-assignment models: Include (i.e., C_ex=∅ ), Exclude (i.e., C_in=∅ ), and Mixed. Horiyama et al. showed that PAU-VC is _2^p -complete on general and NP-complete on bipartite graphs under all three models. This paper shows PAU-VC remains _2^p -complete in all three models even if a given graph is restricted to be planar with maximum degree 3.
The problem is a classic decision problem that is -complete in general graphs, but solvable in linear time and in nondeterministic logspace ( ) for directed acyclic graphs (DAGs). We show that the problem for DAGs lies in ∩ , providing an improved upper bound on its complexity. To the best of our knowledge, this is the first instance where an unambiguous space upper bound has been established for a natural problem without relying on any variant of the classic Reinhardt-Allender double inductive counting technique, which is itself based on the Immerman-Szelepcsényi inductive counting method. Our proof introduces a novel technique, which we call inductive tracing. While reminiscent of inductive counting, it diverges in a key respect: it does not perform any counting across recursive stages. We also apply this routine to obtain a parameterized unambiguous space bound for the problem in DAGs.
A distance- d independent set, for any integer d ≥ 2 , in a simple unweighted graph G is defined as a subset of vertices S such that any two vertices u, v ∈ S are at least distance d apart in G . Given an unweighted graph G and a positive integer k, the Distance- d Independent Set Problem (D d IS) asks to determine whether, there exists a distance- d independent set S in G such that |S| ≥ k . D2IS is equivalent to the well-known classical independent set problem. We denote the maximization version of DdIS as MaxDdIS and the parameterized version as ParaDdIS(k) where the parameter k is the size of a distance-d independent set. In this article, we study the computational complexity of DdIS, MaxDdIS, and ParaDdIS(k) for some popular graph classes: claw-free and bisplit graphs. We have shown that DdIS ( d≥ 3 ) is NP-complete for claw-free graphs with maximum degree 12. Moreover, we prove that DdIS (for d= 3,4 ) is NP-complete for triangle-free and C_6 -free bisplit graphs. Furthermore, we have shown that, for any ϵ > 0 , it is NP-hard to approximate MaxD3IS within a factor n^1/4-ϵ and MaxD4IS within a factor n^1/2-ϵ for triangle-free bisplit graphs. Further, we have shown that ParaD3IS(k) and ParaD4IS(k) are W[1]-hard for triangle-free bisplit graphs.
Contact representations of planar graphs by triangles or – in the bipartite case – by vertical and horizontal segments in the plane can be constructed in linear time. It is also known that in both of these cases, deciding whether a representation of an induced subgraph of an input graph can be extended into a representation of the entire graph is NP-hard. We complement these results by showing that simultaneous representation of two graphs is also NP-hard (both for triangle contact graphs, and for grid contact ones). We show this by a unified reduction via simultaneous contact representations of so-called x-y-stretchable sets of convex sets, which implies NP-hardness of the simultaneous representation problem for many more classes of geometric contact graphs.
We initiate the study of computing diverse triangulations of a given polygon. Given a simple n-gon P, an integer k ≥ 2 , a quality measure σ on the set of triangulations of P and a factor α≥ 1 , we formulate the Diverse and Nice Triangulations (DNT) problem that asks to compute k distinct triangulations T_1,… ,T_k of P such that a) their diversity, ∑ _i < j d(T_i,T_j) , is as large as possible and b) they are nice, i.e., σ (T_i) ≤ασ ^* for all 1≤ i ≤ k . Here, d denotes the symmetric difference of edge sets of two triangulations, and σ ^* denotes the best quality of triangulations of P, e.g., the minimum Euclidean length. As our main result, we provide a poly(n,k) -time approximation algorithm for the DNT problem that returns a collection of k distinct triangulations whose diversity is at least 1 - (1/k) of the optimal, and each triangulation satisfies the quality constraint. This is accomplished by studying bi-criteria triangulations (BCT), which are triangulations that simultaneously optimize two criteria, a topic of independent interest. We complement our approximation algorithms by showing that the DNT problem and the BCT problem are NP-hard. Finally, for the version where diversity is defined as min _i < j d(T_i,T_j) , we show a reduction from the problem of computing optimal Hamming codes, and provide an n^O(k) -time 12 -approximation algorithm. This improves over the naive C_n-2 ()k≈ 2^O(nk) time bound for enumerating all k-tuples among the triangulations of a simple n-gon, where C_n denotes the n-th Catalan number.
