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A complete solution to the biased Alon-Krivelevich-Spencer-Szabó criterion problem for the discrepancy game Scaffolds for Higher Tropical Grassmannians: Foundations Krahn-Szegő type inequalities for graphs $\boldsymbol{2}$-Neighbor Bootstrap Percolation on Odd Graphs Modular inequalities and Alexander polynomials of pencil type conic-line arrangements Counterexamples regarding elementary symmetric partitions Bounds on Multiple $b$-Burst Deletion-Correcting Codes The mex statistic on combinatorial structures Proper Additive Edge Colorings of Regular Graphs Off-diagonal Ramsey numbers Inferring DAGs and Phylogenetic Networks from Least Common Ancestors Additive structures imply more distances in $\mathbb{F}_q^d$ Semi-inducibility of 4-vertex graphs A Structural Condition on Point Sets with Few Distinct Dot Products A quantum N-dimer model Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs Well quasi-order and atomicity for combinatorial structures under consecutive orders The Permanent Rank of a Matrix (Part Three) Note on the Additive Basis Conjecture Complexity of the Zero Set of a Matrix Schubert Ideal The existence of unexpected automorphisms in direct product graphs Reconstruction of oriented matroids from Varchenko-Gelfand algebras Addition theorems for Ziegler pairs of hyperplane arrangements Combinatorial proofs of Petrie Pieri rule and Plethystic Pieri rule Asymptotic structure. III. Excluding a fat tree Asymptotic structure. V. The coarse Menger conjecture in bounded path-width Word-Representable Co-Bipartite Graphs: Vertex Ordering, Representation Number, Speed, and Entropy Borel Kernels in Borel Directed Graphs Large induced subgraphs with prescribed degree parity Maya-Tupi graphs: a generalization of split graphs Recursions, Trains, Trees, and Combinatorial Rod Set Algebra Sufficient minimum degree conditions for the existence of highly connected or edge-connected subgraphs Ramsey numbers for 1-degenerate 3-graphs Bipartite graphs with minimum degree at least 15 are antimagic A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs Bounds on the game isolation number and exact values for paths and cycles Minors of non-hamiltonian polyhedra and the Herschel family Notes on sum-free sets in abelian groups When are Hopf algebras determined by integer sequences? A note on the irrationality of $ζ_2(5)$ Support-sensitive bounds for shortest zero-sum subsequences Weighted Random Dot Product Graphs More efficient sifting for grid norms, and applications to multiparty communication complexity Ordered Yao graphs: maximum degree, edge density, and clique numbers Refined enumeration of planar Eulerian orientations Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent Bubble sort and Howe duality for staircase matrices Self-affine quadrangles Covering half-grids with lines and planes Borel Local Lemma: arbitrary random variables and limited exponential growth Dynamic capacity allocation of hybrid transportation units for cargo-hitching in urban public transportation systems An Efficient Genus Algorithm Based on Graph Rotations Neural Networks and (Virtual) Extended Formulations Some conjectures on $r$-graphs and equivalences Pointwise order of generalized Hofstadter functions G, H and beyond On the periodic decompositions of multidimensional configurations A q-analogue of graph independence polynomials with a group-theoretic interpretation Moment-angle manifolds corresponding to three-dimensional simplicial spheres, chordality and connected sums of products of spheres Strong Embeddings of 3-Connected Cubic Planar Graphs on Surfaces of non-negative Euler Characteristic Higher-rank trees arising from polyhedral graphs $L^p$ averages of the Fourier transform in finite fields A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes Open problems on relations of numerical semigroups Alcove walk models for parabolic Mirković-Vilonen intersections and branching to Levi subgroups A Mathematical Reconstruction of Endothelial Cell Networks Matchings in the hypercube with specified edges Geometric planted matchings beyond the Gaussian model Extensions and Deletions of matroid classes closed under flats String attractors and bi-infinite words Progressive and Rushed Dyck Paths Some results on Kleitman's conjecture Symmetric noncrossing partitions of an annulus with double points On two families of Nekrasov-Okounkov type formulas A remark on certain restricted plane partitions and crystal melting model Lecture hall graphs and the Askey scheme The Partial Partition Complex Generating functions of substitutions Standard Monomials for Positroid Varieties Tiling dense hypergraphs A structural duality for path-decompositions into parts of small radius The grid-minor theorem revisited Sumsets and entropy revisited Structure of cell decompositions in Extremal Szemerédi-Trotter examples Coloring, List Coloring, and Painting Squares of Graphs (and other related problems) Determining a Points Configuration from a Subset of the Pairwise Distances Improved Bounds for Multicovering Hypergraphs Totally Nonnegative Tropical Flags and the Totally Nonnegative Flag Dressian Enumeration of Odd-Dimensional Partitions modulo 4 Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups Multidimensional Manhattan Preferences Balancing permuted copies of multigraphs and integer matrices Mixing times of one-sided $k$-transposition shuffles Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices The pre-Pieri rules Synchronization of strongly connected partial DFAs and prefix codes Large rainbow matchings in edge-colored graphs Surface cluster algebra expansion formulae via loop graphs Lattice structure of Weyl groups via representation theory of preprojective algebras A Cambrian framework for the oriented cycle Combinatorial frameworks for cluster algebras Sortable Elements for Quivers with Cycles
Balanced Domination in Convex Polytopes, Trees, and Grid Graphs
Bojan Nikolic, Marko Djukanovic · 2025-11-10 · via math.CO updates on arXiv.org

This paper addresses two open questions posed in [27] regarding the balanced domination number in graphs. We show that three new classes of graphs, those of convex polytopes A_n, D_n, and Rn'', are d-balanced. Further, we provide a characterization of d-balancedness for rooted trees with two levels of descendants and prove that each full binary tree is d-balanced. Several results for caterpillar graphs are established. Moreover, we determine and prove the exact balanced domination number for grid graphs. Finally, we conclude by providing several open problems of interest.

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