Sprague-Grundy values and complexity for LCTR

Given a Young diagram on $n$ boxes as a non-increasing sequence of integers, we consider the impartial combinatorial game LCTR in which moves consist of removing either the left column or top row of boxes. We show that for both normal and mis\`ere play, the optimal strategy can consist mostly of mirroring the opponent's moves. This allows for computing the Sprague-Grundy value of the given game in $O(\log(n))$ time units, where time unit allows for reading an integer, or performing a basic arithmetic operation. This improves on the previous bound of $O(n)$ time units, due to by Ili\'c (2019), which can be obtained by an improvement of the Sprague-Grundy recursion.Comment: 24 pages, 7 figures, 1 tabl

Some results on LCTR, an impartial game on partitions

We apply the Sprague-Grundy Theorem to LCTR, a new impartial game on partitions in which players take turns removing either the Left Column or the Top Row of the corresponding Young diagram. We establish that the Sprague-Grundy value of any partition is at most $2$, and determine Sprague-Grundy values for several infinite families of partitions. Finally, we devise a dynamic programming approach which, for a given partition $\lambda$ of $n$, determines the corresponding Sprague-Grundy value in $O(n)$ time

Avoidability beyond paths

The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2019 as a common generalization of avoidable vertices and simplicial paths. In 2020, Bonamy, Defrain, Hatzel, and Thiebaut proved that every graph containing an induced path of order $k$ also contains an avoidable induced path of the same order. They also asked whether one could generalize this result to other avoidable structures, leaving the notion of avoidability up to interpretation. In this paper we address this question: we specify the concept of avoidability for arbitrary graphs equipped with two terminal vertices. We provide both positive and negative results, some of which appear to be related to the recent work by Chudnovsky, Norin, Seymour, and Turcotte [arXiv:2301.13175]

Graphs with at most two moplexes

A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: Unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval. In this work we initiate an investigation of $k$-moplex graphs, which are defined as graphs containing at most $k$ moplexes. Of particular interest is the smallest nontrivial case $k=2$, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected $2$-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. On the other hand, we prove that every connected $2$-moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets

Fair allocation of indivisible goods under conflict constraints

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Thereby we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both {\sc Partition} and {\sc Independent Set}. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We can show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).Comment: A preliminary version containing some of the results presented here appeared in the proceedings of IWOCA 2020. Version 3 contains an appendix with a remark about biconvex bipartite graph

Mere središčnosti velikih omrežij

In most networks some edges or vertices are more central than others. To quantify importance of nodes in networks, centrality indices were introduced. For a given structural index, Freeman centralization is a measure of how central a vertex is regarding to how central all the other vertices are with respect to the given index. In the thesis we study several such structural indices like degree, eccentricity, closeness, betweenness centrality, the Wiener index and transmission. We confirm a conjecture by Everett, Sinclair, and Dankelmann regarding the problem of maximizing closeness centralization in two-mode data, where the number of data of each type is fixed. Intuitively, our result states that among all networks obtainable via two-mode data, the largest closeness is achieved by simply locally maximizing the closeness of a node. Mathematically, our study concerns bipartite networks with fixed size bipartitions, and we show that the extremal configuration is a rooted tree of depth ▫$2$▫, where neighbors of the root have an equal or almost equal number of children. We determine the maximum value of eccentricity centralization and (some) maximizing networks for the families of bipartite networks with given partition sizes, tree networks with fixed maximum degree and fixed number of vertices, and networks with fixed number of nodes or edges. As a by-product, we introduce and study a new way of enumerating the nodes of a tree. We also study the centralization of transmission, in particular, we determine the graphs on n vertices which attain the maximum or minimum value. Roughly, the maximizing graphs are comprised of a path which has one end glued to a clique of similar order. The minimizing family of extremal graphs consists of three paths of almost the same length, glued together in one endvertex. Group centrality indices, introduced in 1999 by Borgatti and Everett, measure the importance of sets of nodes in networks. We study the notion of group centralization with respect to eccentricity, degree and betweenness centrality measures. For groups of size ▫$2$▫, we determine the maximum achieved value of group eccentricity and group betweenness centralization and describe the corresponding extremal graphs. For group degree centralization we do the same with arbitrary size of group. For a given integer ▫$k$▫, by reduction to maximum domination problem, we observe that determining the maximum group degree centralization some ▫$k$▫-subset of ▫$V(G)$▫ is ▫${mathcal NP}$▫-hard. We describe polynomial algorithm with the best possible approximation ratio that calculates all centralizations for ▫$1 le k le n$▫ and altogether runs in ▫${mathcal O}(n^2)$▫ time. The constructed algorithm is tested on six real-world networks. In results we observe a property of unimodality of group degree centralization for parameter ▫$k$▫, which may be a new property for studying networks. The well studied Wiener index ▫$W(G)$▫ of a graph ▫$G$▫ is equal to the sum of distances between all pairs of vertices of ▫$G$▫. Denote by ▫$W[{mathcal G}_n]$▫ the set of all values of the Wiener index over all connected graphs on $n$ vertices and let the largest interval which is fully contained in ▫$W[{mathcal G}_n]$▫ be denoted by ▫$W^{rm int}_n$▫. In the thesis, we show that ▫$W^{rm int}_n$▫ is well-defined, it starts at ▫${n choose 2}$▫, and that both ▫$W^{rm int}_n$▫ and ▫$W[{mathcal G}_n]$▫ are of cardinality ▫${1 over 6}n^3 + {mathcal O}(n^2)$▫ (in other words, most of integers between the smallest value ▫${n choose 2}$▫ and the largest value ▫${n+1 choose 3}$▫ are contained in ▫$W^{rm int}_n$▫ and consequently in ▫$W[{mathcal G}_n]$▫.V večini omrežij so nekatera vozlišča ali povezave pomembnejše od drugih. Pomembnost vozlišč v omrežjih lahko izrazimo z merami centralnosti. Podanemu centralnostnemu indeksu lahko določimo indeks Freemanove centralizacije, ki meri relativno centralnost vozlišča v primerjavi s centralnostjno vseh ostalih vozlišč v omrežju. V tej disertaciji analiziramo različne strukturne indekse, kot so stopnja točk, ekscentričnost, centralnost bližine, vmesnostna centralnost, Wienerjev indeks ter totalna razdalja. Potrdimo domnevo avtorjev Everett, Sinclair in Dankelmann glede maksimiziranja bližinske centralizacije v dvodelnih omrežjih, s podanimi velikostmi biparticij. Trdimo, da je največja vrednost centralizacije bližine (med vsemi dvodelnimi omrežji) dosežena, če lokalno maksimiziramo bližinsko centralnost v neki točki. Izkaže se, da je ekstremalna konfiguracija dosežena v korenskem drevesu globine ▫$2$▫, z dodatnim pogojem, da imajo vsi sosedje od korena skoraj enako stopnjo. Med drugim določimo maksimizirajočo vrednost ekscentrične centralizacije ter najdemo nekaj maksimizirajočih omrežij za družine dvodelnih grafov s podanimi velikostmi biparticij, dreves fiksne velikosti s podano maksimalno stopnjo, kot tudi splošnih povezanih omrežij pri podanem številu vozlišč ali povezav. Tekom omenjene analize predstavimo tudi nov način enumeracije drevesnih vozlišč. Totalna razdalja vozlišča ▫$v$▫ je enaka vsoti vseh razdalj med ▫$v$▫ ter vsemi drugimi vozlišči v omrežju. Pri analizi centralizacije totalne razdalje določimo grafe na ▫$n$▫ točkah, ki dosežejo maksimalno ter minimalno vrednost le-tega indeksa. Izkaže se, da so maksimizirajoči grafi sestavljeni iz poti, ki je na enem koncu identificirana s kliko podobne velikosti. Minimizirajoči grafi so sestavljeni iz treh poti podobne velikosti, ki imajo eno krajišče identificirano v skupni točki. Centralnostni indeksi skupin, vpeljani l. 1999 (Everett in Borgatti), merijo pomembnost izbrane množice vozlišč v omrežju. V disertaciji preučujemo skupinske indekse centralizacije ekscentričnosti, stopnje, ter vmesnostne centralnosti. Za skupine velikosti ▫$2$▫ določimo največje dosežene vrednosti skupinske ekscentričnosti ter skupinske vmesnostne centralnosti, hkrati pa določimo tudi pripadajoče ekstremalne grafe. Podobno določimo tudi za skupinsko centralnost stopnje, neodvisno od velikosti skupine. Na problem določanja najboljše skupine v smislu skupinske centralizacije stopnje pri podanem omrežju ▫$G$▫ se osredotočimo tudi algoritmično. Pri podani velikosti skupine k omenjeni problem prevedemo na problem maksimalne kdominacije, ter opazimo da je le-ta ▫${mathcal NP}$▫-težak. Opišemo polinomski algoritem z najboljšim možnim aproksimacijskim koeficientom, ki za vse smiselne velikosti k izračuna centralizacijske vrednosti v skupni časovni zahtevnosti ▫${mathcal O}(n^2)$▫. Omenjeni algoritem testiramo na šestih realnih omrežjih. V rezultatih opazimo lastnost unimodalnosti (za parameter ▫$k$▫), ki se lahko uporabi kot nova metoda za preučevanje velikih omrežij. Wienerjev indeks ▫$W(G)$ grafa $G$▫ je enak vsoti razdalj med vsemi pari vozlišč v ▫$G$▫. Z ▫$W[{mathcal G}_n]$▫ označimo množico vseh vrednosti Wienerjevega indeksa za družino povezanih omrežij na ▫$n$▫ vozliščih, pri čemer največji neprekinjen interval iz ▫$W[{mathcal G}_n]$▫ označimo z ▫$W^{rm int}_n$▫. V disertaciji pokažemo, da je ▫$W^{rm int}_n$▫ smiselno definiran ter se začne v vrednosti ▫${n choose 2}$▫. Poleg tega pokažemo, da je velikost obeh ▫$W^{rm int}_n$▫ ter ▫$W[{mathcal G}_n]$▫ vsaj ▫${1 over 6}n^3 + {mathcal O}(n^2)$▫ tj.v večina vrednosti med ▫${n choose 2}$▫ ter ▫${n+1 choose 3}$▫ je vsebovana v ▫$W^{rm int}_n$▫ (ter posledično tudi v ▫$W[{mathcal G}_n]$▫)