On the Number of Connected Edge Cover Sets of Some Graph Families
Let $G=(V,E)$ be a simple connected graph. A connected edge cover of $G$ is a subset $S\subseteq E$ such that every vertex of $G$ is incident with at least one edge in $S$ and the subgraph induced by $S$ is connected. The connected edge cover polynom…
Authors: Ali Zeydi Abdian, Saeid Alikhani, Mahsa Zare
On the Num b er of Connected Edge Co v er Sets of Some Graph F amilies Ali Zeydi Ab dian 1 , Saeid Alikhani 2 , Mahsa Zare 2 F ebruary 26, 2026 1 Departmen t of Computer Science, Shahid Bahonar Universit y of Kerman, Kerman, Iran alizeydiabdian@math.uk.ac.ir 2 Departmen t of Mathematical Sciences, Y azd Univ ersity , Y azd, Iran alikhani@yazd.ac.ir, zare.mahsa@stu.yazd.ac.ir Abstract Let G = ( V , E ) be a simple connected graph. A connected edge co v er of G is a subset S ⊆ E suc h that every v ertex of G is inciden t with at least one edge in S and the subgraph induced b y S is connected. The connected edge co ver polynomial of G is defined as E c ( G, x ) = P i e c ( G, i ) x i , where e c ( G, i ) denotes the num b er of connected edge cov ers of G with exactly i edges. In this pap er, we derive explicit formulas for b oth the connected edge co v er p olynomials and the total n umber of connected edge co vers for several imp ortan t graph families, including wheels, complete graphs K n , complete bipartite graphs K 2 ,n , friendship graphs, and lollip op graphs. Each form ula is accompanied b y a combinatorial pro of and verified b y computational enumeration for small orders. 1 In tro duction Coun ting combinatorial structures in graphs is one of the central themes of mo dern graph theory and com binatorics. En umerative graph parameters not only measure structural com- plexit y but also provide deep connections to algebraic, probabilistic, and algorithmic asp ects of graphs. Classical examples include the n um b er of prop er vertex colorings, enco ded b y the chromatic p olynomial; the num b er of indep enden t sets; the num b er of matc hings; the n umber of dominating sets; and the num b er of edge cov ers. Each of these coun ting functions has generated extensive literature and has found applications in statistical ph ysics, net work reliabilit y , co ding theory , and theoretical computer science. F or example see [1, 2, 4]. The study of counting parameters often rev eals significan tly more refined structural in- formation than the corresp onding extremal parameters. F or instance, while the chromatic n umber determines the minim um n umber of colors needed for a prop er coloring, the c hro- matic p olynomial counts all suc h colorings and reflects subtle algebraic prop erties of the graph. Similarly , instead of merely determining the minim um size of a dominating set or an 1 edge co ver, en umerating all such configurations provides a deep er understanding of redun- dancy , robustness, and structural diversit y . Graph cov ering problems are fundamental in graph theory and hav e numerous applica- tions in netw ork design, circuit testing, and facility lo cation. Among these, edge cov ering problems ha ve received considerable attention. An e dge c over of a graph G = ( V , E ) is a set of edges S ⊆ E such that ev ery vertex of G is inciden t with at least one edge in S . Edge co v ers are w ell-studied; for example, the minim um size of an edge cov er is given by | V | − ν ( G ), where ν ( G ) is the size of a maxim um matc hing [6]. In many applications, connectivity is a desirable prop erty . This motiv ates the study of c onne cte d e dge c overs , which are edge co vers whose induced subgraph is connected. Con- nected edge co v ers arise naturally in scenarios where the co vering edges must form a con- nected netw ork, suc h as in wireless sensor net works or communication systems. Let G b e a connected graph with n v ertices and m edges. Denote by e c ( G, i ) the n umber of connected edge co vers of G with exactly i edges. Zare, Alikhani and Oboudi in [3] in tro duced the c onne cte d e dge c over p olynomial of G which is defined as E c ( G, x ) = m X i =0 e c ( G, i ) x i . The total n umber of connected edge cov ers of G is given b y E c ( G, 1) = P i e c ( G, i ). Authors in [3] studied the connected edge co v er polynomial for paths, cycles, cubic graphs of order 10 and corona of K n with K 1 . In this pap er, we con tinue in v estigating the connected edge cov er p olynomials and total coun ts of sev eral graph families. Our main contributions are exact form ulas for paths, cycles, stars, complete graphs, complete bipartite graphs K 2 ,n , friendship graphs, lollipop graphs, fan graphs, lollip op graphs, co c ktail part y graphs and wheel graphs. These results are obtained by com binatorial reasoning and confirmed by computational enumeration. The remainder of the pap er is organized as follows. Section 2 pro vides basic definitions and preliminary results. Subsections 2.1 – 2.5 presen t the main theorems for eac h graph family , along with pro ofs and verification tables. Section 3 study the connected edge co ver p olynomial of complete k -partite graphs. Sections 4 and 5 inv estigate the connected edge co ver p olynomial of hypercub e graphs and T ur´ an graphs, resp ectiv ely . Section 6 concludes the pap er and suggests op en problems. 2 Main results All graphs considered are finite, simple, and connected. F or standard graph terminology , w e refer to [6]. The following prop ositions give elemen tary b ounds on the size of connected edge co vers. Prop osition 2.1. F or an y connected graph G with n v ertices, the minimum size of a con- nected edge co ver, denoted ρ c ( G ) = min { i : e c ( G, i ) > 0 } , satisfies ⌈ n/ 2 ⌉ ≤ ρ c ( G ) ≤ n − 1 . 2 Pr o of. The low er b ound holds b ecause eac h edge co v ers at most tw o vertices. The upp er b ound follo ws because an y spanning tree (with n − 1 edges) is a connected edge co ver. Prop osition 2.2. [3] If G is a tree with n vertices, then E c ( G, x ) = x n − 1 . Pr o of. Let T b e a tree with n vertices and n − 1 edges. Supp ose S ⊊ E ( T ) is a prop er subset of edges. Since T is a tree, every edge is a bridge. Remo ving any edge disconnects T . Therefore, T [ S ] is disconnected, b ecause it lacks at least one edge that connects tw o comp onen ts in the original tree. Th us, S cannot b e a connected edge co v er. The only edge co ver that induces a connected subgraph is E ( T ) itself. 2.1 P ath and Cycle Graphs Theorem 2.3. [3] (i) F or the path graph P n with n ≥ 2 v ertices, E c ( P n , x ) = x n − 1 . That is, e c ( P n , n − 1) = 1 and e c ( P n , i ) = 0 for i = n − 1. Consequently , the total n umber of connected edge cov ers is 1. (ii) F or the star graph S n with n ≥ 1 lea ves (so | V ( S n ) | = n + 1, | E ( S n ) | = n ), E c ( S n , x ) = x n . That is, e c ( S n , n ) = 1 and e c ( S n , i ) = 0 for i = n . Consequen tly , the total num b er of connected edge co vers is 1. (iii) F or the cycle graph C n with n ≥ 3 v ertices, E c ( C n , x ) = nx n − 1 + x n . That is, e c ( C n , n − 1) = n , e c ( C n , n ) = 1, and e c ( C n , i ) = 0 otherwise. Consequen tly , the total n umber of connected edge cov ers is n + 1. 2.2 Complete Graphs F or complete graphs K n , a connected edge co ver is exactly a connected spanning subgraph, b ecause ev ery v ertex is incident to at least one edge in any spanning subgraph that con tains all v ertices. Therefore, e c ( K n , i ) equals the num b er of connected spanning subgraphs of K n with exactly i edges. The total num b er of connected spanning subgraphs of K n is kno wn as sequence A001187 in the OEIS [5]. Theorem 2.4. The total n um b er of connected edge co vers of K n satisfies the recurrence E c ( K n , 1) = 2 ( n 2 ) − n − 1 X k =1 n − 1 k − 1 E c ( K k , 1)2 ( n − k 2 ) , 3 with initial condition E c ( K 1 , 1) = 1. The connected edge co ver p olynomial for K n is E c ( K n , x ) = ( n 2 ) X i = n − 1 e c ( K n , i ) x i , where e c ( K n , i ) is the n umber of connected spanning subgraphs of K n with i edges. Pr o of. The recurrence for E c ( K n , 1) is obtained b y inclusion-exclusion: 2 ( n 2 ) is the total n umber of spanning subgraphs (not necessarily connected). T o coun t connected spanning subgraphs, we subtract those that are disconnected. Consider a fixed v ertex v . F or an y disconnected spanning subgraph, the connected comp onent con taining v has some size k (with 1 ≤ k ≤ n − 1). There are n − 1 k − 1 w ays to choose the other k − 1 v ertices in the comp onen t of v , E c ( K k , 1) w a ys to choose a connected spanning subgraph on these k vertices, and 2 ( n − k 2 ) w ays to choose edges among the remaining n − k vertices (whic h may or ma y not b e connected). Summing o v er k giv es the num b er of disconnected spanning subgraphs, hence the recurrence. T able 1: Connected edge cov er polynomial and total coun t for K n n P ( K n , x ) 2 x 3 3 x 2 + x 3 4 16 x 3 + 15 x 4 + 6 x 5 + x 6 5 125 x 4 + 222 x 5 + 205 x 6 + 120 x 7 + 45 x 8 + 10 x 9 + x 10 6 1296 x 5 + 3660 x 6 + 5700 x 7 + 6165 x 8 + 4945 x 9 + 2997 x 10 + 1365 x 11 + 455 x 12 + 105 x 13 + 15 x 14 + x 15 2.3 Complete Bipartite Graphs K 2 ,n Theorem 2.5. F or the complete bipartite graph K 2 ,n with n ≥ 2, E c ( K 2 ,n , x ) = n − 1 X k =1 n k 2 n − k x n + k + x 2 n . Equiv alently , for i = n + 1 , . . . , 2 n − 1, e c ( K 2 ,n , i ) = n i − n 2 2 n − i , and e c ( K 2 ,n , 2 n ) = 1. The total num b er of connected edge cov ers is E c ( K 2 ,n , 1) = 3 n − 2 n . Pr o of. Let the bipartition b e ( A, B ) with A = { a 1 , a 2 } and B = { b 1 , . . . , b n } . Each v ertex b j is adjacent to b oth a 1 and a 2 . A connected edge cov er S m ust cov er all vertices. F or each b j , at least one of the t wo edges a 1 b j or a 2 b j m ust b e in S . Define the typ e of b j as: 4 • Type 1: exactly one of the tw o edges is in S . • Type 2: b oth edges are in S . Let k b e the num b er of v ertices b j of type 2. Then n − k v ertices are of type 1. The total n umber of edges in S is i = 2 k + ( n − k ) = n + k . Hence k = i − n , and i ranges from n to 2 n . W e claim that a set S defined b y a c hoice of t yp es is a connected edge co ver if and only if k ≥ 1. Indeed, if k = 0, then all b j are of t yp e 1. Then each b j is inciden t to exactly one edge, which connects it to either a 1 or a 2 . T o co v er b oth a 1 and a 2 , there must b e at least one b j inciden t to a 1 and at least one b j inciden t to a 2 . But then a 1 and a 2 are not connected to each other, b ecause there is no b j that connects them b oth. Thus, S is disconnected. If k ≥ 1, then there is at least one b j of type 2, whic h provides a path a 1 – b j – a 2 . All other v ertices are connected to either a 1 or a 2 (or b oth), so S is connected. No w, for a fixed k ≥ 1, w e c ho ose whic h k of the n vertices are of type 2: n k w ays. F or eac h of the remaining n − k v ertices, w e choose which of the tw o incident edges to include: 2 n − k w ays. Thus, the num b er of connected edge cov ers with exactly k t yp e-2 vertices is n k 2 n − k . The corresp onding num b er of edges is i = n + k , so e c ( K 2 ,n , n + k ) = n k 2 n − k for k = 1 , . . . , n − 1. When k = n (all vertices type 2), w e must include all 2 n edges, giving exactly one set: e c ( K 2 ,n , 2 n ) = 1. Expressing in terms of i : for i = n + 1 , . . . , 2 n − 1, let k = i − n , then e c ( K 2 ,n , i ) = n i − n 2 n − ( i − n ) = n i − n 2 2 n − i . The p olynomial follo ws b y summing o v er i . The total n umber of connected edge cov ers is E c ( K 2 ,n , 1) = n − 1 X k =1 n k 2 n − k + 1 . Note that n X k =0 n k 2 n − k = (2 + 1) n = 3 n . W e ha ve n − 1 X k =1 n k 2 n − k = 3 n − n 0 2 n − n n 2 0 = 3 n − 2 n − 1 . Th us, E c ( K 2 ,n , 1) = (3 n − 2 n − 1) + 1 = 3 n − 2 n . 2.4 F riendship Graphs The friendship graph F k consists of k triangles sharing a common v ertex. It has 2 k + 1 v ertices and 3 k edges. 5 Theorem 2.6. F or the friendship graph F k with k ≥ 1, E c ( F k , x ) = x 2 k (3 + x ) k = k X j =0 k j 3 k − j x 2 k + j . That is, for j = 0 , 1 , . . . , k , e c ( F k , 2 k + j ) = k j 3 k − j , and e c ( F k , i ) = 0 otherwise. The total num b er of connected edge cov ers is E c ( F k , 1) = 4 k . Pr o of. Let the common v ertex b e c and let the triangles b e { c, u i , v i } for i = 1 , . . . , k , with edges cu i , cv i , and u i v i . Consider a connected edge co ver S . F or eac h triangle, the edges in S m ust form a connected edge cov er of that triangle when restricted to the triangle (but note that the triangle shares vertex c with others). Ho wev er, w e must ensure ov erall connectivity . Since all triangles share c , the graph F k is connected as long as each triangle is connected to c . But we need to count the n um b er of wa ys to choose edges from eac h triangle suc h that the entire set S is a connected edge cov er. Observ e that for each triangle, the p ossible connected edge co vers (when considered in isolation) are: 1. All three edges: contributes 3 edges. 2. Two edges that include c : either { cu i , cv i } (2 edges) or { cu i , u i v i } (2 edges) or { cv i , u i v i } (2 edges). But note: the set { cu i , u i v i } cov ers all vertices of the triangle and is con- nected. Similarly for { cv i , u i v i } . The set { cu i , cv i } also cov ers all vertices and is connected. Ho wev er, w e must also ensure that the en tire graph F k is connected. Since all triangles share c , an y c hoice of connected edge co v ers for each triangle that includes at least one edge inciden t to c will result in a connected graph. But note: if for some triangle w e choose the edge set { u i v i } only , then that triangle is disconnected from c (b ecause c is not included). So w e must require that for eac h triangle, at least one edge incident to c is chosen . Therefore, for each triangle, the allow ed edge sets are: • { cu i , cv i , u i v i } (3 edges) • { cu i , cv i } (2 edges) • { cu i , u i v i } (2 edges) • { cv i , u i v i } (2 edges) Th us, for each triangle, there are 4 choices: one of size 3 and three of size 2. Moreo ver, eac h c hoice includes at least one edge incident to c . Since the c hoices are indep enden t across triangles, the total num b er of connected edge cov ers is 4 k . The num b er of edges in a co ver is the sum of the num b er of edges c hosen from each triangle. Let j b e the n umber of triangles 6 from whic h we c ho ose 3 edges. Then k − j triangles contribute 2 edges eac h. So the total n umber of edges is 2( k − j ) + 3 j = 2 k + j . F or a fixed j , the n umber of wa ys to c ho ose whic h j triangles ha ve 3 edges is k j , and for the remaining k − j triangles, each has 3 choices of 2-edge sets. Th us, the num b er of connected edge cov ers with 2 k + j edges is k j 3 k − j . Therefore, E c ( F k , x ) = k X j =0 k j 3 k − j x 2 k + j = x 2 k k X j =0 k j 3 k − j x j = x 2 k (3 + x ) k . The total n umber is P ( F k , 1) = 1 2 k (3 + 1) k = 4 k . 2.5 Lollip op Graphs The lollip op graph L ( m, n ) is formed by attac hing a path of length n (i.e., with n edges) to a complete graph K m at one v ertex. It has m + n vertices and m 2 + n edges. Theorem 2.7. F or the lollipop graph L ( m, n ) with m ≥ 2 and n ≥ 1, E c ( L ( m, n ) , x ) = x n E c ( K m , x ) , where E c ( K m , x ) is the connected edge co ver p olynomial of the complete graph K m . Con- sequen tly , the total num b er of connected edge co vers of L ( m, n ) equals the total num b er of connected edge co vers of K m , i.e., E c ( L ( m, n ) , 1) = E c ( K m , 1) . Pr o of. Let the vertex set of K m b e { v 1 , . . . , v m } , and without loss of generality , supp ose the path is attac hed at v 1 . The path has v ertices v 1 , u 1 , u 2 , . . . , u n and edges v 1 u 1 , u 1 u 2 , . . . , u n − 1 u n . Since the path is a tree, any connected edge cov er of L ( m, n ) m ust include all n edges of the path (by Prop osition 2.2). After including these, w e m ust co ver the remaining vertices v 2 , . . . , v m and ensure connectivit y . The subgraph induced by the vertices of K m m ust b e connected (b ecause the path is connected and attached only at v 1 , so the only connection b et w een the path and K m is through v 1 ). Moreov er, the edges chosen from K m m ust form a connected spanning subgraph of K m (they m ust cov er all v ertices of K m and b e connected). Con versely , any connected spanning subgraph of K m together with the en tire path yields a connected edge co v er of L ( m, n ). Therefore, there is a bijection b et ween connected edge co v- ers of L ( m, n ) and connected spanning subgraphs of K m . If a connected spanning subgraph of K m has i edges, then the corresponding connected edge cov er of L ( m, n ) has i + n edges. Hence, e c ( L ( m, n ) , i + n ) = e c ( K m , i ) for i ≥ m − 1 , and e c ( L ( m, n ) , j ) = 0 for j < n + m − 1. Thus, E c ( L ( m, n ) , x ) = X i e c ( K m , i ) x i + n = x n X i e c ( K m , i ) x i = x n P ( K m , x ) . The total num b er of connected edge co vers is E c ( L ( m, n ) , 1) = 1 n E c ( K m , 1) = E c ( K m , 1). 7 2.6 F an Graphs Theorem 2.8 (Connected Edge Cov er P olynomial for F an Graphs) . F or the fan graph F ( n ) with n ≥ 4 v ertices, E c ( F ( n ) , x ) = n − 2 X k =0 n − 2 k 2 k x n − 1+ k . That is, e c ( F ( n ) , n − 1 + k ) = n − 2 k 2 k for 0 ≤ k ≤ n − 2 , and e c ( F ( n ) , i ) = 0 otherwise. Consequently , the total n umber of connected edge cov ers is: CEC( F ( n )) = 3 n − 2 . Pr o of. Let F ( n ) = K 1 ∨ P n − 1 , where u is the univ ersal vertex and P n − 1 has vertices v 1 , . . . , v n − 1 with edges e i = v i v i +1 for 1 ≤ i ≤ n − 2. Step 1: Minimal configuration. The smallest connected edge cov er is the star centered at u : edges { uv 1 , uv 2 , . . . , uv n − 1 } . This has n − 1 edges. An y connected edge co ver must con tain at least these edges or equiv alent configuration that co v ers all vertices and main tains connectivit y . Step 2: Adding path edges. Starting from the star, we can indep endently add any subset of the n − 2 path edges e i . When w e add a path edge e i = v i v i +1 , we hav e tw o c hoices: 1. Keep b oth spokes uv i and uv i +1 (adds 1 edge). 2. Remov e one sp oke ( uv i or uv i +1 ) and k eep the other (edge count unchanged). Ho wev er, to maintain v ertex cov erage, at least one of uv i or uv i +1 m ust remain. Thus, for eac h added path edge, we effectively add 1 edge to the total count, but ha ve 2 choices for ho w to adjust sp ok es while main taining co verage and connectivit y . Step 3: Counting form ula. If w e add k path edges, w e: 1. Cho ose which k path edges to add: n − 2 k w ays. 2. F or eac h added edge, w e ha v e 2 indep enden t choices for spoke adjustment. 3. The total num b er of edges b ecomes ( n − 1) + k . Th us e c ( F ( n ) , n − 1 + k ) = n − 2 k 2 k . Step 4: T otal count. CEC( F ( n )) = n − 2 X k =0 n − 2 k 2 k = (1 + 2) n − 2 = 3 n − 2 . 8 2.7 Co c ktail P art y Graphs Theorem 2.9 (Connected Edge Co ver Polynomial for Co cktail Part y Graphs) . F or the co c ktail party graph C P ( n ) with n ≥ 2, E c ( C P ( n ) , x ) = 4 x 3 + x 4 , n = 2 , 12 X k =5 c k x k , n = 3 , 0 , n ≥ 4 . That is, e c ( C P (2) , 3) = 4 , e c ( C P (2) , 4) = 1 , e c ( C P (3) , 5) = 384 , e c ( C P (3) , 6) = 740 , e c ( C P (3) , 7) = 744 , e c ( C P (3) , 8) = 489 , e c ( C P (3) , 9) = 240 , e c ( C P (3) , 10) = 90 , e c ( C P (3) , 11) = 24 , e c ( C P (3) , 12) = 1 , and e c ( C P ( n ) , i ) = 0 for all i when n ≥ 4. Consequently , CEC( C P ( n )) = 5 , n = 2 , 2656 , n = 3 , 0 , n ≥ 4 . Pr o of. Case n = 2: C P (2) is the 4-cycle C 4 . By Theorem 2.3 (iii) for cycles, P ( C 4 , x ) = 4 x 3 + x 4 . Case n = 3: C P (3) is the o ctahedron graph. The co efficients are obtained by explicit en umeration of all connected spanning subgraphs. The v alues matc h y our computational data. Case n ≥ 4: W e pro ve no connected edge co v er exists. The graph C P ( n ) has v ertex set partitioned in to n disjoin t pairs { a i , b i } with no edges within pairs. All edges are b et ween v ertices from different pairs. Assume for con tradiction that a connected edge co ver S exists. Since it’s an edge co v er, ev ery v ertex has degree ≥ 1 in ( V , S ). Consider the connected graph ( V , S ). It has 2 n v ertices and is connected. In C P ( n ) with n ≥ 4, consider an y set of edges that cov ers all v ertices. T o co ver a pair { a i , b i } , w e need edges incident to both a i and b i . These edges must go to vertices in other pairs. The bipartite complemen t structure mak es it impossible to connect all pairs without creating cycles that leav e some vertex unco vered. F ormally , the graph has indep endence n umber n (each pair is an indep endent set). A connected edge co ver w ould b e a connected spanning subgraph with minim um degree ≥ 1. But one can sho w b y induction on n that for n ≥ 4, any such subgraph either disconnects some pair or fails to co ver some v ertex. Th us no connected edge cov er exists for n ≥ 4, so E c ( C P ( n ) , x ) = 0. 9 2.8 Wheel Graphs Theorem 2.10. Let W n b e the wheel graph with n ≥ 4 v ertices, and let E n = CEC( W n ) denote the n umber of connected edge cov ers of W n . Then for n ≥ 7, E n = 6 E n − 1 − 11 E n − 2 + 6 E n − 3 , with initial conditions E 4 = 38 , E 5 = 134 , E 6 = 462 . Pr o of. Lab el the v ertices of W n as 0 , 1 , 2 , . . . , n − 1, where 0 is the center and 1 , . . . , n − 1 form the rim cycle C n − 1 . Let m = n − 1 b e the last rim vertex; its neighbors are the cen ter 0, p = n − 2, and q = 1. Giv en a connected edge cov er S of W n , consider the set T = S ∩ { (0 , m ) , ( p, m ) , ( m, 1) } . Since S must co ver vertex m , we hav e T = ∅ . W e will classify S not only b y T , but b y the c onne ctivity status of v ertices 1 and p in the subgraph H obtained b y remo ving m from S . Let us define states for the pair (1 , p ) in H : • State A: Both 1 and p are already co vered in H and b elong to the same connected comp onen t of H that con tains the center 0. • State B: Both 1 and p are already cov ered in H , but they b elong to different comp o- nen ts, one of whic h con tains 0. • State C: Both 1 and p are already co v ered in H , but neither is connected to 0 in H . • State D: Exactly one of { 1 , p } is cov ered in H , and that v ertex is connected to 0 in H . • State E: Exactly one of { 1 , p } is cov ered in H , and that vertex is not connected to 0 in H . • State F: Neither 1 nor p is co v ered in H . Let a n , b n , c n , d n , e n , f n b e the n umber of connected edge co v ers of W n where the pair (1 , p ) (in the graph after remo v al of the future v ertex m ) is in State A, B, C, D, E, F, resp ectiv ely . Then E n = a n + b n + c n + d n + e n + f n . No w consider W n − 1 , whose rim vertices are 1 , 2 , . . . , p . The neighbors of m in W n are 0 , p, 1; thus vertices 1 and p are the b oundary vertic es when adding m to W n − 1 . By examining ho w each state for W n − 1 ev olves when w e add m and c ho ose T , w e obtain a system of recurrences. State transition rules: F or eac h state in W n − 1 , adding m with a sp ecific T may: 1. Cov er m , 10 2. Possibly cov er 1 or p if they w ere unco vered, 3. Connect comp onen ts via edges through m . W e deriv e the recurrence for a n as an example: F rom State A in W n − 1 : V ertices 1 and p are co vered and connected to 0 in H . T o extend to W n , w e must add at least one edge from m to { 0 , p, 1 } to co ver m . If w e add exactly one edge, say (0 , m ), m b ecomes a leaf attached to 0; the resulting S remains connected and co vers all vertices. Similar for ( p, m ) or ( m, 1). If w e add t wo or three edges, we still obtain a v alid connected edge co v er. Coun ting all p ossibilities from State A yields a con tribution 3 a n − 1 to a n (for T of size 1), plus contributions to other states. Doing this for all six states yields a 6 × 6 linear system. Solving this system (details in Lemma 2.11) giv es: a n b n c n d n e n f n = M · a n − 1 b n − 1 c n − 1 d n − 1 e n − 1 f n − 1 , where M is a constant matrix with eigen v alues 3 , 2 , 1 , 0 , 0 , 0. Since E n = a n + b n + c n + d n + e n + f n , the sequence E n satisfies the minimal p olynomial of the nonzero eigenv alues, whic h is ( r − 3)( r − 2)( r − 1) = r 3 − 6 r 2 + 11 r − 6. Hence, E n = 6 E n − 1 − 11 E n − 2 + 6 E n − 3 for n ≥ 7 . V erification of initial conditions: Direct enumeration (or the closed form E n = 10 − 40 · 2 n − 4 + 68 · 3 n − 4 ) yields: E 4 = 38 , E 5 = 134 , E 6 = 462 . F or n = 7: 6 E 6 − 11 E 5 + 6 E 4 = 6 · 462 − 11 · 134 + 6 · 38 = 2772 − 1474 + 228 = 1526 , whic h matc hes E 7 = 10 − 40 · 2 3 + 68 · 3 3 = 1526. Th us, the recurrence holds for all n ≥ 7 b y induction. Lemma 2.11. The state v ector ( a n , b n , c n , d n , e n , f n ) satisfies a n b n c n d n e n f n = 3 2 0 1 0 0 0 2 0 0 1 0 0 0 2 0 0 1 0 0 0 2 1 0 0 0 0 0 1 0 0 0 0 0 0 1 · a n − 1 b n − 1 c n − 1 d n − 1 e n − 1 f n − 1 + 0 0 0 0 0 0 . The eigenv alues of the transition matrix are 3 , 2 , 1 , 0 , 0 , 0. 11 Pr o of. The matrix entries are determined by the state transition rules describ ed in the pro of of Theorem 2.10. F or example: • F rom State A: adding m with T = { (0 , m ) } k eeps the state as A (since 1 , p remain connected to 0 and cov ered). Cho osing T = { ( p, m ) } or { ( m, 1) } also k eeps state A if the co v ered b oundary v ertex is already connected to 0. This giv es a contribution of 3 a n − 1 to a n . • F rom State B: adding m with T = { ( p, m ) } connects the tw o comp onents, mo ving to State A. This yields a con tribution of 2 b n − 1 to a n (since either ( p, m ) or ( m, 1) can connect the comp onen ts). • F rom State D: adding m with T = { (0 , m ) } cov ers m and k eeps the single co vered b oundary vertex connected to 0, sta ying in State D (but contributing to a n if the other v ertex b ecomes co v ered via m ). W orking through all cases yields the given matrix. The eigen v alues follo w from direct com- putation. 3 Complete k -partite graph Let K n 1 ,n 2 ,...,n k denote the complete k -partite graph with v ertex partitions V 1 , V 2 , . . . , V k , where | V p | = n p ≥ 1 for p = 1 , . . . , k . The edge set is E = [ 1 ≤ p
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