Delaunay graphs have been used in CAD/CAM, sensor networks, and geographic information systems. We investigate the reliability properties of nodes in Delaunay graphs. For measuring the reliability, we formulate the concept of roaming-region for nodes. The \({roaming-region}\) \( R(i) \) of a Delaunay node \( v_i \) is such that the Delaunay graph does not change as long as \( v_i \) remains within \( R(i) \). A node \( v_i \) with a large roaming region \( R(i) \) such that \( v_i \) is positioned near the center of \( R(i) \) is identified as a reliable node. Two types of roaming regions called (i) \({lateral\; roaming\; region}\) \( LR(i) \) and (ii) \({radial\; roaming\; region}\) \( RR(i) \) are distinguished to develop the algorithm. The roaming region itself is expressed as the intersection of \( RR(i) \) and \( LR(i) \). For nodes inside the convex hull, called \({deep\; internal\; nodes}\), we present an \( O(n^2) \) time algorithm for computing their roaming region, where \( n \) is the number of nodes in the Delaunay triangulation. We finally discuss generalization and extension of the proposed algorithm.

A pencyclic graph on \( v \) vertices is called pancyclic if it contains cycles of every length from \( 3 \) to \( v \). In this paper we address the question: what is the minimum number of edges in a pancyclic graph? We present a simple analysis using chord patterns.

Software interaction test suites serve two complementary roles. They are employed to systematically verify that, for some strength \( t \), no \( t \)-way interaction of a system’s parameters causes a fault. They are also employed to locate a faulty configuration when at least one interaction fault remains. Algorithms to find such test suites employing a number of tests close to the minimum have been extensively explored, in order to test all \( t \)-way interactions. However, when faults remain, the expected number of tests needed to reveal an interaction fault is also important. One might anticipate that the test suites of minimum size also have the lowest expected time to detection of an interaction fault; or, at the very least, that some test suite of minimum size does. However, in this paper it is shown that minimum test suite size and lowest expected time to fault detection are incompatible objectives. This underlies a challenging problem of how to generate test suites that have early coverage of \( t \)-way interactions, in order to reduce time to fault detection. A hybrid approach is developed that combines a simple greedy algorithm with heuristic search to construct one test at a time while attempting to maximize the number of \( t \)-way interactions covered by the earliest tests.

Faults in software systems often occur due to interactions between parameters. Several studies show that faults are caused by 2-way through 6-way interactions of parameters. In the context of test suite prioritization, we have studied prioritization by 2-way inter-window interaction coverage and found that this criterion is effective at finding faults quickly in the test execution cycle. However, since faults may be caused by interactions between more than 2 parameters, in this paper, we provide a greedy algorithm for test suite prioritization by \( n \)-way combinatorial coverage of inter-window interactions. While greedy algorithms that generate Combinatorial Interaction Test suites enumerate and track the coverage of all possible \( t \)-tuples and constraints, we have noticed that our user-session-based test suites often do not contain every possible \( t \)-tuple, and we can take advantage of this in our algorithm by only storing \( t \)-tuples that appear in the test suite. Our empirical study shows both time and memory usage associated with our algorithm for 3-way inter-window parameter-value interaction coverage. Further, we conduct an empirical study where we compare 2-way and 3-way combinatorial coverage of inter-window parameter interactions in terms of the rate of fault detection for a web application called Schoolmate and a user-session-based test suite. Our results show that the rate of fault detection for 2-way and 3-way prioritization are within \(1\%\) of each other, but 2-way provides a slightly better result. A closer look at the characteristics of the web application, test cases, and faults reveals that most faults are triggered by 2-way interactions. We motivate the need for future work to examine a larger set of empirical studies to identify characteristics of web applications that benefit from prioritization with higher strength inter-window event interaction coverage.

In this work, we present a greedy algorithm for covering the set of incomplete STRIPS planning domain interpretations by \( t \)-strength diagnoses. We present a greedy algorithm to cover the incomplete domain model interpretations with a set of plans by iteratively generating plans so that each additional plan is biased to cover at least one new interpretation not previously covered. We also present a second greedy algorithm to construct a set of plans that covers all \( t \)-strength diagnoses of plan failure for plans in the incomplete domain model. We show that covering domain interpretations by \( t \)-strength diagnoses leads to increased coverage by a set of plans despite potentially lower coverage per plan because covering by \( t \)-strength diagnoses leads to a more scalable approach to planning where more plans can be found.

The article presents the compatibility matrix method and illustrates it with the application to the \( \text{P} \) vs \( \text{NP} \) problem. The method is a generalization of descriptive geometry: in the method, we draft problems and solve them utilizing the image creation technique. The method reveals: \( \text{P} = \text{NP} = \text{PSPACE} \subseteq \text{P/poly} \), etc.

Our previous paper [9] applied a lopsided version of the Lovász Local Lemma that allows negative dependency graphs [5] to the space of random matchings in \( K_{2n} \), deriving new proofs to a number of results on the enumeration of regular graphs with excluded cycles through the configuration model [3]. Here we extend this from excluded cycles to some excluded balanced subgraphs, and derive asymptotic results on the probability that a random regular multigraph from the configuration model contains at least one from a family of balanced subgraphs in question.

The induced path number \( \rho(G) \) of a graph \( G \) is defined as the minimum number of subsets into which the vertex set of \( G \) can be partitioned so that each subset induces a path. A Nordhaus-Gaddum type result is a (tight) lower or upper bound on the sum (or product) of a parameter of a graph and its complement. If \( G \) is a subgraph of \( H \), then the graph \( H – E(G) \) is the complement of \( G \) relative to \( H \). In this paper, we consider Nordhaus-Gaddum type results for the parameter \( \rho \) when the relative complement is taken with respect to the complete bipartite graph \( K_{m,n} \).

Rado constructed a (simple) denumerable graph \( R \) with the positive integers as vertex set with the following edges: For given \( m \) and \( n \) with \( m < n \), \( m \) is adjacent to \( n \) if \( n \) has a \( 1 \) in the \( m \)'th position of its binary expansion. It is well known that \( R \) is a universal graph in the set \( \mathcal{I} \) of all countable graphs (since every graph in \( \mathcal{I} \) is isomorphic to an induced subgraph of \( R \)) and that \( R \) can be characterized using this notion and that of being homogeneous and having the extension property. In this paper, we extend these notions to arbitrary induced-hereditary properties (of graphs), relate them to the construction of a universal graph for any such property, and obtain results which remind one of some characterizations of \( R \).

In this note, we prove that for any tree \( T \), \( \gamma_{\leq2}(T) \leq \gamma_\gamma(T) \leq ir(T) \leq \gamma(T) \), where \( \gamma_{\leq2}(G) \) is the distance-2 domination number, \( ir(T) \) is the (lower) irredundance number, \( \gamma(T) \) is the domination number, and \( \gamma_\gamma(T) \), newly defined here, equals the minimum cardinality of a set of vertices that dominates a minimum dominating set of \( T \).