Let H = (V, E) be a hypergraph with n vertices and m edges. A subset S subset of V is an eliminating vertex feedback set of H if H/S contains no cycles, and we denote its minimum cardinality by D-c(H). Analogously, a subset A subset of E is called an eliminating edge feedback set of H if H/A contains no cycles, and its minimum cardinality is denoted by D '(c)(H). In this paper, we establish upper bounds for both parameters. For the vertex version, we prove that D-c(H) < (k-1)n-m + 1 for any k-regular connected hypergraph H. For the edge version, we show that: (i) D '(c)(H) < n/2 for any hypergraph H; (ii) D '(c)(H) < n/4 for any 2-regular hypergraph H with girth at least 4; and (iii) D '(c)(H) < n/3 for any 2-regular or 3-regular hypergraph H with girth at least 3. Furthermore, we characterize the extremal regular hypergraphs attaining these upper bounds. Based on our proofs, we also design combinatorial algorithms for computing the eliminating feedback numbers. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We propose and study the diversity-aware l-centrum (div-l-centrum) problem, where we are given a set of clients and a set of facilities in a metric space, integers k and l, and r[i] (i is an element of [t]) for each facility group. The diversity constraints refer to the requirement that each group must include at least r[i] open facilities. The goal is to find a subset of facilities of size at most k so as to minimize the sum of the l largest distances from a client to its nearest open facility, while satisfying the diversity constraints. As our main contribution, we provide a fixed-parameter tractable (1 + 2/e + epsilon)-approximation algorithm that improves upon the previous (3 + epsilon) for the div-l-centrum problem, based on a combination of leader selection, coreset construction and submodular optimization techniques. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
For a graph G = (V , E), let G(n) denote the graph whose vertex set is V n in which two distinct vertices (u(1), u(2),..., u(n)) and (v(1), v(2),..., v(n)) are adjacent, if and only if for all 1 <= i <= n, either u(i) = v(i) or u(i )v(i) is an element of E. The Shannon capacity of G is defined to be c(G) = sup(n) alpha(G(n)) (1/n) , where alpha(G(n)) is the maximum size of an independent set of vertices in G(n). In this paper, we prove that alpha(G boxed times H) = alpha(G)alpha(H) for any connected graph G and connected bipartite graph H. We prove that for any graph G and bipartite graph H (or point-symmetric H which satisfies that alpha(H)omega(H) =|V (H)|), if the Shannon capacity of G is finite, then the Shannon capacity of G boxed times H is finite. The r-color Ramsey number R(k(1),...,k(r)) is defined as the smallest positive integer N such that in every edge-coloring of K (N) by r colors there is a monochromatic copy of Kki in color i for some 1 <= i <= k. In 2002, Robertson obtained a lower bound on the r-color Ramsey number R(k(1),k(2),...,k(r)). We establish a new lower bound on the r-color Ramsey number of graphs. From this, we can improve the theorem of Robertson for some cases. A specific result improves the inequality R(7, 9, 7, 7) >= 5384 to that R(7, 9, 7, 7) >= 64261. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Given an undirected graph G and a set A⊆V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G)∖A. An A-path is called an (A,ℓ)-path if the length of the path is exactly ℓ. In the (A,ℓ)-Path Packing problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A,ℓ)-paths in G or not. The problem is already known to be fixed-parameter tractable (FPT) when parameterized by k+ℓ using color coding while it remains Para-NP-hard when parameterized by k (Hamiltonian Path) or ℓ (P3-Partition) alone. Therefore, a logical direction to pursue this problem is to examine it in relation to structural parameters. Belmonte et al. [Algorithmica 2022] initiated a study along these lines and proved that ALPP parameterized by pw+|A| is W[1]-hard where pw is the pathwidth of the input graph. In this paper, we strengthen their result and prove that it is unlikely that ALPP is fixed-parameter tractable even with respect to a bigger parameter (|A|+dtp) where dtp denotes the distance from the input graph to a path graph (distance to path). Following this, we consider the parameters cvd+|A| and cvd+ℓ and provide FPT algorithms with respect to these parameters. Finally, when parameterized by vc, the vertex cover number of the input graph, we provide an O(vc2) vertex kernel for (A,ℓ)-Path Packing.
We study the Labeled Contractibility problem, where the input consists of two vertex-labeled graphs G and H, and the goal is to determine whether H can be obtained from G via a sequence of edge contractions. Lafond and Marchand [WADS 2025] initiated the parameterized complexity study of this problem, showing it to be W[1]-hard when parameterized by the number k of allowed contractions. They also proved that the problem is fixed-parameter tractable when parameterized by the tree-width tw of G, via an application of Courcelle's theorem resulting in a non-constructive algorithm. In this work, we present a constructive fixed-parameter algorithm for Labeled Contractibility with running time 2(O(tw2)) center dot vertical bar V (G)vertical bar(O(1)). We also prove that unless the Exponential Time Hypothesis (ETH) fails, it does not admit an algorithm running in time 2(o(tw2)) center dot vertical bar V (G)vertical bar(O(1)). This result adds Labeled Contractibility to a small list of problems that admit such a lower bound and matching algorithm. We further strengthen existing hardness results by showing that the problem remains NPcomplete even when both input graphs have bounded maximum degree. We also investigate parameterizations by ( k + delta(G)) where delta(G) denotes the degeneracy of G, and rule out the existence of subexponential-time algorithms. This answers question raised in Lafond and Marchand [WADS 2025]. We additionally provide an improved FPT algorithm with better dependence on ( k + delta(G)) than previously known. Finally, we analyze a brute-force algorithm for Labeled Contractibility with running time vertical bar V (H)vertical bar(O(vertical bar V (G)vertical bar)), and show that this running time is optimal under ETH.
