We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the 1-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in ℝⁿ described by O(n) linear inequalities is hard to approximate within a factor of (1-ε)ln n for any constant ε > 0, unless P = NP. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.

In the Coloring Reconfiguration problem, we are given two proper k-colorings of a graph and asked to decide whether one can be transformed into the other by repeatedly applying a specified recoloring rule, while maintaining a proper coloring throughout. For this problem, two recoloring rules have been widely studied: single-vertex recoloring and Kempe chain recoloring. In this paper, we introduce a new rule, called color swapping, where two adjacent vertices may exchange their colors, so that the resulting coloring remains proper, and study the computational complexity of the problem under this rule. We first establish a complexity dichotomy with respect to k: the problem is solvable in polynomial time for k ≤ 2, and is PSPACE-complete for k ≥ 3. We further show that the problem remains PSPACE-complete even on restricted graph classes, including bipartite graphs, split graphs, and planar graphs of bounded degree. In contrast, we present polynomial-time algorithms for several graph classes: for paths when k = 3, for split graphs when k is fixed, and for cographs when k is arbitrary.

In reconfiguration problems, we are given two feasible solutions to a graph problem and asked whether one can be transformed into the other via a sequence of feasible intermediate solutions under a given reconfiguration rule. While earlier work focused on modifying a single element at a time, recent studies have started examining how different rules impact computational complexity. Motivated by recent progress, we study Independent Set Reconfiguration (ISR) and Vertex Cover Reconfiguration (VCR) under the k-Token Jumping (k-TJ) and k-Token Sliding (k-TS) models. In k-TJ, up to k vertices may be replaced, while k-TS additionally requires a perfect matching between removed and added vertices. It is known that the complexity of ISR crucially depends on k, ranging from PSPACE-complete and NP-complete to polynomial-time solvable. In this paper, we further explore the gradient of computational complexity of the problems. We first show that ISR under k-TJ with k = |I| - μ remains NP-hard when μ is any fixed positive integer and the input graph is restricted to graphs of maximum degree 3 or planar graphs of maximum degree 4, where |I| is the size of feasible solutions. In addition, we prove that the problem belongs to NP not only for μ = O(1) but also for μ = O(log |I|). In contrast, we show that VCR under k-TJ is in XP when parameterized by μ = |S| - k, where |S| is the size of feasible solutions. Furthermore, we establish the PSPACE-completeness of ISR and VCR under both k-TJ and k-TS on several graph classes, for fixed k as well as superconstant k relative to the size of feasible solutions.

In a reconfiguration problem, we are given two feasible solutions of a combinatorial problem and our goal is to determine whether it is possible to reconfigure one into the other, with the steps dictated by specific reconfiguration rules. Traditionally, most studies on reconfiguration problems have focused on rules that allow changing a single element at a time. In contrast, this paper considers scenarios in which k ≥ 2 elements can be changed simultaneously. We investigate the general reconfiguration problem of isomorphisms. For the Induced Subgraph Isomorphism Reconfiguration problem, we show that the problem remains PSPACE -complete even under stringent constraints on the pattern graph when k is constant. We then give two meta-theorems applicable when k is slightly less than the number of vertices in the pattern graph. In addition, we investigate the complexity of the Independent Set Reconfiguration problem, which is a special case of the Induced Subgraph Isomorphism Reconfiguration problem.

Consider a graph G where each vertex has a threshold. A vertex v in G is activated if the number of active vertices adjacent to v is at least as many as its threshold. A vertex subset A 0 of G is a target set if eventually all vertices in G are activated by initially activating vertices of A 0 . The Target Set Selection problem ( TSS ) involves finding a smallest target set of G . This problem has already been extensively studied and is known to be NP-hard even for very restricted conditions. In this paper, we analyze TSS and its weighted variant, called the Weighted Target Set Selection problem ( WTSS ), from the perspective of parameterized complexity. Let k be the solution size and let ℓ be the maximum threshold. We first show that TSS is W[1]-hard for split graphs when parameterized by k + ℓ , and W[2]-hard for cographs when parameterized by k . We next prove that WTSS is W[2]-hard for trivially perfect graphs when parameterized by k . On the other hand, we show that WTSS can be solved in O ( n log ⁡ n ) time for complete graphs with n vertices. Additionally, we design FPT algorithms for WTSS when parameterized by nd + ℓ , tw + ℓ , ce , and vc , where nd , tw , ce , and vc are the neighborhood diversity, the treewidth, the cluster editing number, and the vertex cover number of the input graph, respectively.

Let Π₁, Π₂, …, Π_c be graph properties for a fixed integer c. Then, (Π₁, Π₂, …, Π_c)-Partition is the problem of asking whether the vertex set of a given graph can be partitioned into c subsets V₁, V₂, …, V_c such that the subgraph induced by V_i satisfies the graph property Π_i for every i ∈ {1,2, …, c}. Minimization and parameterized variants of (Π₁, Π₂, …, Π_c)-Partition have been studied for several specific graph properties, where the size of the vertex subset V₁ satisfying Π₁ is minimized or taken as a parameter. In this paper, we first show that the minimization variant is hard to approximate for any nontrivial additive hereditary graph properties, unless c = 2 and both Π₁ and Π₂ are classes of edgeless graphs. We then give FPT algorithms for the parameterized variant when restricted to the case where c = 2, Π₁ is a hereditary graph property, and Π₂ is the class of acyclic graphs.