We study almost envy-freeness in house allocation, where m houses are to be allocated among n agents so that every agent receives exactly one house. An envy-free allocation need not exist, and therefore we may have to settle for relaxations. We study different aggregate measures of envy as markers of fairness. In particular, we define the amount of envy experienced by an agent a w.r.t. an allocation to be the number of agents that agent a envies under that allocation. We quantify the envy generated by an allocation using three different metrics: 1) the number of agents who are envious; 2) the maximum amount of envy experienced by any agent; and 3) the total amount of envy experienced by all agents, and look for allocations that minimize one of the three metrics. We prove a host of algorithmic and hardness results. We also suggest practical approaches for these problems via integer linear program (ILP) formulations and report the findings of our experimental evaluation of ILPs. Finally, we study the price of fairness, which quantifies the loss of welfare we must suffer due to the fairness requirements, and present tight bounds as well as algorithms that simultaneously optimize both welfare and fairness.

The eternal vertex cover game is played between an attacker and a defender on an undirected graph G. The defender identifies k vertices to position guards initially. The attacker, on their turn, attacks an edge e, and the defender must move a guard along e to defend the attack. The defender may move other guards as well, under the constraint that every guard moves at most once and to a neighboring vertex. The smallest number of guards required to defend attacks forever is called the eternal vertex cover number of G, denoted evc(G). For any graph G, evc(G) is at least mvc(G) (the vertex cover number of G). A graph is Spartan if evc(G) = mvc(G). It is known that a bipartite graph is Spartan if and only if every edge belongs to a perfect matching. We show that the only König graphs that are Spartan are the bipartite Spartan graphs. We also give new lower bounds for evc(G), generalizing a known lower bound based on cut vertices. We finally show a new matching-based characterization of all Spartan graphs.

We study almost envy-freeness in house allocation, where m houses are to be allocated among n agents so that every agent receives exactly one house. An envy-free allocation need not exist, and therefore we may have to settle for relaxations. We study different aggregate measures of envy as markers of fairness. In particular, we define the amount of envy experienced by an agent a w.r.t. an allocation to be the number of agents that agent a envies under that allocation. We quantify the envy generated by an allocation using three different metrics: 1) the number of agents who are envious; 2) the maximum amount of envy experienced by any agent; and 3) the total amount of envy experienced by all agents, and look for allocations that minimize one of the three metrics. We prove a host of algorithmic and hardness results. We also suggest practical approaches for these problems via integer linear program (ILP) formulations and report the findings of our experimental evaluation of ILPs. Finally, we study the price of fairness, which quantifies the loss of welfare we must suffer due to the fairness requirements, and present tight bounds as well as algorithms that simultaneously optimize both welfare and fairness.

We study the Boolean Satisfiability problem (SAT) in the framework of diversity, where one asks for multiple solutions that are mutually far apart (i.e., sufficiently dissimilar from each other) for a suitable notion of distance/dissimilarity between solutions. Interpreting assignments as bit vectors, we take their Hamming distance to quantify dissimilarity, and we focus on the problem of finding two solutions. Specifically, we define the problem Max Differ SAT (resp. Exact Differ SAT) as follows: Given a Boolean formula ϕ on n variables, decide whether ϕ has two satisfying assignments that differ on at least (resp. exactly) d variables. We study the classical and parameterized (in parameters d and n-d) complexities of Max Differ SAT and Exact Differ SAT, when restricted to some classes of formulas on which SAT is known to be polynomial-time solvable. In particular, we consider affine formulas, Krom formulas (i.e., 2-CNF formulas) and hitting formulas. For affine formulas, we show the following: Both problems are polynomial-time solvable when each equation has at most two variables. Exact Differ SAT is NP-hard, even when each equation has at most three variables and each variable appears in at most four equations. Also, Max Differ SAT is NP-hard, even when each equation has at most four variables. Both problems are 𝖶[1]-hard in the parameter n-d. In contrast, when parameterized by d, Exact Differ SAT is 𝖶[1]-hard, but Max Differ SAT admits a single-exponential FPT algorithm and a polynomial-kernel. For Krom formulas, we show the following: Both problems are polynomial-time solvable when each variable appears in at most two clauses. Also, both problems are 𝖶[1]-hard in the parameter d (and therefore, it turns out, also NP-hard), even on monotone inputs (i.e., formulas with no negative literals). Finally, for hitting formulas, we show that both problems can be solved in polynomial-time.

For any fixed positive integer r and a given budget k, the r-Eigenvalue Vertex Deletion (r-EVD) problem asks if a graph G admits a subset S of at most k vertices such that the adjacency matrix of G⧵S has at most r distinct eigenvalues. The edge deletion, edge addition, and edge editing variants are defined analogously. For r = 1, r-EVD is equivalent to the Vertex Cover problem. For r = 2, it turns out that r-EVD amounts to removing a subset S of at most k vertices so that G⧵ S is a cluster graph where all connected components have the same size. We show that r-EVD is NP-complete even on bipartite graphs with maximum degree four for every fixed r > 2, and FPT when parameterized by the solution size and the maximum degree of the graph. We also establish several results for the special case when r = 2. For the vertex deletion variant, we show that 2-EVD is NP-complete even on triangle-free and 3d-regular graphs for any d ≥ 2, and also NP-complete on d-regular graphs for any d ≥ 8. The edge deletion, addition, and editing variants are all NP-complete for r = 2. The edge deletion problem admits a polynomial time algorithm if the input is a cluster graph, while - in contrast - the edge addition variant is hard even when the input is a cluster graph. We show that the edge addition variant has a quadratic kernel. The edge deletion and vertex deletion variants admit a single-exponential FPT algorithm when parameterized by the solution size alone. Our main contribution is to develop the complexity landscape for the problem of modifying a graph with the aim of reducing the number of distinct eigenvalues in the spectrum of its adjacency matrix. It turns out that this captures, apart from Vertex Cover, also a natural variation of the problem of modifying to a cluster graph as a special case, which we believe may be of independent interest.

We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph $G$ and is denoted by $evc(G)$. It is clear that $evc(G)$ is at least $mvc(G)$, the size of a minimum vertex cover of $G$. We say that $G$ is Spartan if $evc(G) = mvc(G)$. The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on $2n$ vertices where every edge belongs to a perfect matching, an easy strategy is to have $n$ guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs.

The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of $k \ge 1$ agents. While the initial positions of Facilitator's agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator's goal to bring both her agents at same vertex and Divider's goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with $k$ agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents. This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when parameterized by the modular width and the number of allowed rounds. However, they left open the complexity of the problem from the perspective of other structural parameters. In particular, they explicitly asked whether the problem admits an FPT or XP-algorithm with respect to the treewidth of the input graph. We answer this question in the negative and show that Rendezvous is co-NP-hard even for graphs of constant treewidth. Further, we show that the problem is co-W[1]-hard when parameterized by the feedback vertex set number and the number of agents, and is unlikely to admit a polynomial kernel when parameterized by the vertex cover number and the number of agents. Complementing these hardness results, we show that the Rendezvous is FPT when parameterized by both the vertex cover number and the solution size. Finally, for graphs of treewidth at most two and girds, we show that the problem can be solved in polynomial time.

We study the parameterized complexity of the optimal defense and optimal attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a set of votes) and two integers k_a and k_d corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the optimal defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most k_d voter groups such that, no matter which k_a voter groups the attacker attacks, the out-come of the election does not change. In the optimal attack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking k_a voter groups such that, no matter which k_d voter groups the defender defends, the outcome of the election is always different from the original (without any attack) one. We show that both the optimal defense problem and the optimal attack problem are computationally intractable for every scoring rule and the Condorcet voting rule even when we have only3candidates. We also show that the optimal defense problem for every scoring rule and the Condorcet voting rule is W[2]-hard for both the parameters k_a and k_d, while it admits a fixed parameter tractable algorithm parameterized by the combined parameter (ka, kd). The optimal attack problem for every scoring rule and the Condorcet voting rule turns out to be much harder – it is W[1]-hard even for the combined parameter (ka, kd). We propose two greedy algorithms for the OPTIMAL DEFENSE problem and empirically show that they perform effectively on reasonable voting profiles.

The Chamberlin-Courant and Monroe rules are fundamental and well-studied rules in the literature of multi-winner elections. The problem of determining if there exists a committee of size k that has a Chamberlin-Courant (respectively, Monroe) dissatisfaction score of at most r is known to be NP-complete. We consider the following natural problems in this setting: a) given a committee S of size k as input, is it an optimal k-sized committee?, and b) given a candidate c and a committee size k, does there exist an optimal k-sized committee that contains c? In this work, we resolve the complexity of both problems for the Chamberlin-Courant and Monroe voting rules in the settings of rankings as well as approval ballots. We show that verifying if a given committee is optimal is coNP-complete whilst the latter problem is complete for Theta_2^P. Our contribution fills an essential gap in the literature for these important multi-winner rules.

We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be placed on yet unburnt vertices on alternate rounds to block the fire. In this work, we consider the problem of saving a critical subset of nodes from catching fire, given a total budget on the number of firefighters. We show that the problem is para-NP-hard when parameterized by the size of the critical set. We also show that it is fixed-parameter tractable on general graphs when parameterized by the number of firefighters. We also demonstrate improved running times on trees and establish that the problem is unlikely to admit a polynomial kernelization (even when restricted to trees). Our work is the first to exploit the connection between the firefighting problem and the notions of important separators and tight separator sequences. Finally, we consider the spreading model of the firefighting game, a closely related problem, and show that the problem of saving a critical set parameterized by the number of firefighters is W[2]-hard, which contrasts our FPT result for the non-spreading model.

This thesis is in the area called computational social choice which is an intersection area of algorithms and social choice theory.

In this paper we extend the classical notion of strong and weak backdoor sets for SAT and CSP by allowing that different instantiations of the backdoor variables result in instances that belong to different base classes; the union of the base classes forms a heterogeneous base class. Backdoor sets to heterogeneous base classes can be much smaller than backdoor sets to homogeneous ones, hence they are much more desirable but possibly harder to find. We draw a detailed complexity landscape for the problem of detecting strong and weak backdoor sets into heterogeneous base classes for SAT and CSP.

Bribery in elections is an important problem in computational social choice theory. We introduce and study two important special cases of the bribery problem, namely, FRUGAL-BRIBERY and FRUGAL-$BRIBERY where the briber is frugal in nature. By this, we mean that the briber is only able to influence voters who benefit from the suggestion of the briber. More formally, a voter is vulnerable if the outcome of the election improves according to her own preference when she accepts the suggestion of the briber. In the FRUGAL-BRIBERY problem, the goal is to make a certain candidate win the election by changing only the vulnerable votes. In the FRUGAL-$BRIBERY problem, the vulnerable votes have prices and the goal is to make a certain candidate win the election by changing only the vulnerable votes, subject to a budget constraint. We show that both the FRUGAL-BRIBERY and the FRUGAL-$BRIBERY problems are intractable for many commonly used voting rules for weighted as well as unweighted elections. These intractability results demonstrate that bribery is a hard computational problem, in the sense that several special cases of this problem continue to be computationally intractable. This strengthens the view that bribery, although a possible attack on an election in principle, may be infeasible in practice.

Policy Iteration (PI) (Howard 1960) is a classical method for computing an optimal policy for a finite Markov Decision Problem (MDP). The method is conceptually simple: starting from some initial policy, “policy improvement” is repeatedly performed to obtain progressively dominating policies, until eventually, an optimal policy is reached. Being remarkably efficient in practice, PI is often favoured over alternative approaches such as Value Iteration and Linear Programming. Unfortunately, even after several decades of study, theoretical bounds on the complexity of PI remain unsatisfactory. For an MDP with n states and k actions, Mansour and Singh (1999) bound the number of iterations taken by Howard’s PI, the canonical variant of the method, by O(kn / n). This bound merely improves upon the trivial bound of kn by a linear factor. However, a randomised variant of PI introduced by Mansour and Singh (1999) does yield an exponential improvement, with its expected number of iterations bounded by O(((1 + 2/log2(k)) k / 2)n).With the objective of furnishing improved upper bounds for PI, we introduce two randomised procedures in this paper. Our first contribution is a routine to find a good initial policy for PI. After evaluating a number of randomly generated policies, this procedure applies a novel criterion to pick one to initialise PI. When PI is subsequently applied, we show that the expected number of policy evaluations—including both the initialisation and the improvement stages—remains bounded in expectation by O(kn/2). The key construction employed in this routine is a total order on the set of policies. Our second contribution is a randomised action-switching rule for PI, which admits a bound of O((2 + ln(k – 1))n) on the expected number of iterations. To the best of our knowledge, this is the tightest complexity bound known for PI when k >= 3.

We study a general class of problems called $\mathcal{F}$-Deletion problems. In an $\mathcal{F}$-Deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family ${\cal F}$ of forbidden minors. We study the problem parameterized by $k$, using $p$-$\mathcal{F}$-Deletion to refer to the parameterized version of the problem. We obtain a number of algorithmic results on the $p$-$\mathcal{F}$-Deletion problem when $\mathcal{F}$ contains a planar graph. We give a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leaves, as an induced subgraph, where $t$ is a fixed integer and an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when $\cal F$ only contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set, and Diamond Hitting Set. The generic kernelization algorithm is based on a nontrivial application of protrusion techniques, previously used only for problems on topological graph classes.

A backdoor set of a CNF formula is a set of variables such that fixing the truth values of the variables from this set moves the formula into a polynomial-time de-cidable class. In this work we obtain several algorithmic results for solving d-SAT, by exploiting backdoors to d-CNF formulas whose incidence graphs have small treewidth. For a CNF formula ϕ and integer t, a strong backdoor set to treewidth t is a set of variables such that each possible partial assignment τ to this set reduces ϕ to a formula whose incidence graph is of treewidth at most t. A weak backdoor set to treewidth t is a set of variables such that there is a partial assignment to this set that reduces ϕ to a satisfiable formula of treewidth at most t. Our main contribution is an algorithm that, given a d-CNF formula ϕ and an integer k, in time , either finds a satisfying assignment of ϕ, or reports correctly that ϕ is not satisfiable, or concludes correctly that ϕ has no weak or strong backdoor set to treewidth t of size at most k. As a consequence of the above, we show that d-SAT parameterized by the size of a smallest weak/strong backdoor set to formulas of treewidth t, is fixed-parameter tractable. Prior to our work, such results were know only for the very special case of t = 1 (Gaspers and Szeider, ICALP 2012). Our result not only extends the previous work, it also improves the running time substantially. The running time of our algorithm is linear in the input size for every fixed k. Moreover, the exponential dependence on the parameter k is asymptotically optimal under Exponential Time Hypothesis (ETH). One of our main technical contributions is a linear time “protrusion replacer” improving over a (n log2 n)-time procedure of Fomin et al. (FOCS 2012). The new deterministic linear time protrusion replacer has several applications in kernelization and parameterized algorithms.

In this work, we study the well-known $r$-Dimensional $k$-Matching ($(r,k)$-DM), and $r$-Set $k$-Packing ($(r,k)$-SP) problems. Given a universe $U := U_1 \uplus \cdots \uplus U_r$ and an $r$-uniform family $\mathcal{F} \subseteq U_1 \times \cdots \times U_r$, the $(r,k)$-DM problem asks if $\mathcal{F}$ admits a collection of $k$ mutually disjoint sets. Given a universe $U$ and an $r$-uniform family $\mathcal{F}\subseteq 2^U$, the $(r,k)$-SP problem asks if $\mathcal{F}$ admits a collection of $k$ mutually disjoint sets. We employ techniques based on dynamic programming and representative families. This leads to a deterministic algorithm with running time $\mathcal{O} (2.851^{(r-1)k}\cdot|\mathcal{F}|\cdot n\log^2 n\cdot \log W)$ for the weighted version of $(r,k)$-DM, where $W$ is the maximum weight in the input, and a deterministic algorithm with running time $\mathcal{O}(2.851^{(r-0.5501)k}\cdot|\mathcal{F}|\cdot n\log^2 n\cdot \log W)$ for the weighted version of $(r,k)$-SP. Thus, we significantly improve the previous best known deterministic running times for $(r,k)$-DM and $(r,k)$-SP and the previous best known running times for their weighted versions. We rely on structural properties of $(r,k)$-DM and $(r,k)$-SP to develop algorithms that are faster than those that can be obtained by a standard use of representative sets. Incorporating the principles of iterative expansion, we obtain a better algorithm for $(3,k)$-DM, running in time $\mathcal{O}(2.004^{3k}\cdot|\mathcal{F}| \cdot n\log^2 n)$. We believe that this algorithm demonstrates an interesting application of representative families in conjunction with more traditional techniques. Furthermore, we present kernels of size $\mathcal{O}(e^rr(k-1)^r\log W)$ for the weighted versions of $(r,k)$-DM and $(r,k)$-SP, improving the previous best known kernels of size $\mathcal{O}(r!r(k-1)^r\log W)$ for these problems.

Backdoor sets represent clever reasoning shortcuts through the search space for SAT and CSP. By instantiating the backdoor variables one reduces the given instance to several easy instances that belong to a tractable class.The overall time needed to solve the instance is exponential in the size of the backdoor set, hence it is a challenging problem to find a small backdoor set if one exists; over the last years this problem has been subject of intensive research. In this paper we extend the classical notion of a strong backdoor set by allowing that different instantiations of the backdoor variables result in instances that belong to different base classes; the union of the base classes forms a heterogeneous base class. Backdoor sets to heterogeneous base classes can be much smaller than backdoor sets to homogeneous ones, hence they are much more desirable but possibly harder to find. We draw a detailed complexity landscape for the problem of detecting strong backdoor sets into heterogeneous base classes for SAT and CSP. We provide algorithms that establish fixed-parameter tractability under natural parameterizations, and we contrast the tractability results with hardness results that pinpoint the theoretical limits. Our results apply to the current state-of-the-art of tractable classes of CSP and SAT that are definable by restricting the constraint language.

Given a universe U := U_1 + .... + U_r and a r-uniform family F which is a subset of U_1 x .... x U_r, the r-dimensional matching problem asks if F admits a collection of k mutually disjoint sets. The special case when r=3 is the classic 3-Dimensional Matching problem. Recently, several improvements have been suggested for these (and closely related) problems in the setting of randomized parameterized algorithms. Also, many approaches have evolved for deterministic parameterized algorithms. For instance, for the 3-Dimensional Matching problem, a combination of color coding and iterative expansion yields a running time of O^*(2.80^{(3k)}), and for the r-dimensional matching problem, a recently developed derandomization for known algebraic techniques leads to a running time of O^*(5.44^{(r-1)k}). In this work, we employ techniques based on dynamic programming and representative families, leading to a deterministic algorithm with running time O^*(2.85^{(r-1)k}) for the r-Dimensional Matching problem. Further, we incorporate the principles of iterative expansion used in the literature [TALG 2012] to obtain a better algorithm for 3D-matching, with a running time of O^*(2.003^{(3k)}). Apart from the significantly improved running times, we believe that these algorithms demonstrate an interesting application of representative families in conjunction with more traditional techniques.

Let F be a finite set of graphs. In the F-DELETION problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. F-DELETION is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as VERTEX COVER, FEEDBACK VERTEX SET or TREEWIDTH η-DELETION. In this paper we obtain a number of generic algorithmic results about F-DELETION, when F contains at least one planar graph. The highlights of our work are · A constant factor approximation algorithm for the optimization version of F-DELETION; · A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2O(k)n), for the parameterized version of F-DELETION where all graphs in F are connected; · A polynomial kernel for parameterized F-DELETION. These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results - constant factor approximation, polynomial kernelization and FPT algorithms - are stringed together by a common theme of polynomial time preprocessing.

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $θ_c$ as a minor for a fixed integer $c$. The graph $θ_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.

In the Connected Dominating Set problem we are given as input a graph $G$ and a positive integer $k$, and are asked if there is a set $S$ of at most $k$ vertices of $G$ such that $S$ is a dominating set of $G$ and the subgraph induced by $S$ is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set. Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer $k$ (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function $g(k)$. The new instance is called a $g(k)$ kernel for the problem. If $g(k)$ is a polynomial in $k$ then we say that the problem admits polynomial kernels. The girth of a graph $G$ is the length of a shortest cycle in $G$. It turns out that Connected Dominating Set is ``hard'' on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth $3$ or $4$ (since the problem is W[2]-hard); (b) admits a $g(k)$ kernel, where $g(k)$ is $k^{O(k)}$, on graphs of girth $5$ or $6$ but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic ($O(k^3)$) kernel on graphs of girth at least $7$. While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs.